L(s) = 1 | + 4·3-s + 8·9-s + 80·17-s − 16·19-s + 36·25-s − 60·27-s − 104·41-s − 140·43-s + 320·51-s − 64·57-s + 224·59-s − 60·67-s − 264·73-s + 144·75-s − 372·81-s − 628·83-s + 408·97-s − 860·107-s − 216·113-s + 472·121-s − 416·123-s + 127-s − 560·129-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
L(s) = 1 | + 4/3·3-s + 8/9·9-s + 4.70·17-s − 0.842·19-s + 1.43·25-s − 2.22·27-s − 2.53·41-s − 3.25·43-s + 6.27·51-s − 1.12·57-s + 3.79·59-s − 0.895·67-s − 3.61·73-s + 1.91·75-s − 4.59·81-s − 7.56·83-s + 4.20·97-s − 8.03·107-s − 1.91·113-s + 3.90·121-s − 3.38·123-s + 0.00787·127-s − 4.34·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(16.42457363\) |
\(L(\frac12)\) |
\(\approx\) |
\(16.42457363\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 36 T^{2} + 68 p T^{4} + 18852 T^{6} - 4914 p^{3} T^{8} + 18852 p^{4} T^{10} + 68 p^{9} T^{12} - 36 p^{12} T^{14} + p^{16} T^{16} \) |
good | 3 | \( ( 1 - 2 T + 2 T^{2} + 34 T^{3} - 20 T^{4} - 10 T^{5} + 638 T^{6} + 1994 T^{7} - 3482 T^{8} + 1994 p^{2} T^{9} + 638 p^{4} T^{10} - 10 p^{6} T^{11} - 20 p^{8} T^{12} + 34 p^{10} T^{13} + 2 p^{12} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 7 | \( 1 - 3820 T^{4} + 1702492 p T^{8} - 35880522964 T^{12} + 114370840901494 T^{16} - 35880522964 p^{8} T^{20} + 1702492 p^{17} T^{24} - 3820 p^{24} T^{28} + p^{32} T^{32} \) |
| 11 | \( ( 1 - 236 T^{2} + 25348 T^{4} - 4668308 T^{6} + 802174774 T^{8} - 4668308 p^{4} T^{10} + 25348 p^{8} T^{12} - 236 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 13 | \( 1 - 57832 T^{4} + 1163317468 T^{8} - 49532428306264 T^{12} + 2329202974370476870 T^{16} - 49532428306264 p^{8} T^{20} + 1163317468 p^{16} T^{24} - 57832 p^{24} T^{28} + p^{32} T^{32} \) |
| 17 | \( ( 1 - 40 T + 800 T^{2} - 14360 T^{3} + 188732 T^{4} - 837320 T^{5} - 14388000 T^{6} + 899521800 T^{7} - 23809845626 T^{8} + 899521800 p^{2} T^{9} - 14388000 p^{4} T^{10} - 837320 p^{6} T^{11} + 188732 p^{8} T^{12} - 14360 p^{10} T^{13} + 800 p^{12} T^{14} - 40 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 + 4 T + 652 T^{2} - 5060 T^{3} + 193606 T^{4} - 5060 p^{2} T^{5} + 652 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 23 | \( 1 + 436436 T^{4} - 1686913148 T^{8} - 9640341122092180 T^{12} + \)\(25\!\cdots\!06\)\( T^{16} - 9640341122092180 p^{8} T^{20} - 1686913148 p^{16} T^{24} + 436436 p^{24} T^{28} + p^{32} T^{32} \) |
| 29 | \( ( 1 + 1544 T^{2} + 1604668 T^{4} + 707686712 T^{6} + 480608903494 T^{8} + 707686712 p^{4} T^{10} + 1604668 p^{8} T^{12} + 1544 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 31 | \( ( 1 + 4236 T^{2} + 8282932 T^{4} + 10501818420 T^{6} + 10707341577750 T^{8} + 10501818420 p^{4} T^{10} + 8282932 p^{8} T^{12} + 4236 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 37 | \( 1 - 1651048 T^{4} + 640175610076 T^{8} + 8784630165623383592 T^{12} - \)\(14\!\cdots\!26\)\( T^{16} + 8784630165623383592 p^{8} T^{20} + 640175610076 p^{16} T^{24} - 1651048 p^{24} T^{28} + p^{32} T^{32} \) |
| 41 | \( ( 1 + 26 T + 5668 T^{2} + 114518 T^{3} + 13714198 T^{4} + 114518 p^{2} T^{5} + 5668 p^{4} T^{6} + 26 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 43 | \( ( 1 + 70 T + 2450 T^{2} + 77450 T^{3} + 4403180 T^{4} + 7609610 p T^{5} + 15116386350 T^{6} + 697968108450 T^{7} + 32526450776038 T^{8} + 697968108450 p^{2} T^{9} + 15116386350 p^{4} T^{10} + 7609610 p^{7} T^{11} + 4403180 p^{8} T^{12} + 77450 p^{10} T^{13} + 2450 p^{12} T^{14} + 70 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 47 | \( 1 - 26388652 T^{4} + 353454337736836 T^{8} - \)\(29\!\cdots\!32\)\( T^{12} + \)\(17\!\cdots\!18\)\( T^{16} - \)\(29\!\cdots\!32\)\( p^{8} T^{20} + 353454337736836 p^{16} T^{24} - 26388652 p^{24} T^{28} + p^{32} T^{32} \) |
| 53 | \( 1 + 639512 T^{4} - 82732457712164 T^{8} - 18644811461010919768 T^{12} + \)\(28\!\cdots\!34\)\( T^{16} - 18644811461010919768 p^{8} T^{20} - 82732457712164 p^{16} T^{24} + 639512 p^{24} T^{28} + p^{32} T^{32} \) |
| 59 | \( ( 1 - 56 T + 188 p T^{2} - 601160 T^{3} + 53022646 T^{4} - 601160 p^{2} T^{5} + 188 p^{5} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 61 | \( ( 1 - 22356 T^{2} + 236482036 T^{4} - 1551728773548 T^{6} + 6920371161099990 T^{8} - 1551728773548 p^{4} T^{10} + 236482036 p^{8} T^{12} - 22356 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 67 | \( ( 1 + 30 T + 450 T^{2} + 31890 T^{3} - 21668372 T^{4} - 252866970 T^{5} + 2673244350 T^{6} + 1969823885130 T^{7} + 809157113842278 T^{8} + 1969823885130 p^{2} T^{9} + 2673244350 p^{4} T^{10} - 252866970 p^{6} T^{11} - 21668372 p^{8} T^{12} + 31890 p^{10} T^{13} + 450 p^{12} T^{14} + 30 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 + 22524 T^{2} + 221602900 T^{4} + 1330398616644 T^{6} + 6630039003829590 T^{8} + 1330398616644 p^{4} T^{10} + 221602900 p^{8} T^{12} + 22524 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 73 | \( ( 1 + 132 T + 8712 T^{2} + 738156 T^{3} + 30316876 T^{4} - 690176988 T^{5} - 82786845960 T^{6} - 11668804283028 T^{7} - 1345563598434330 T^{8} - 11668804283028 p^{2} T^{9} - 82786845960 p^{4} T^{10} - 690176988 p^{6} T^{11} + 30316876 p^{8} T^{12} + 738156 p^{10} T^{13} + 8712 p^{12} T^{14} + 132 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 - 29352 T^{2} + 450221020 T^{4} - 4569485596056 T^{6} + 33373106783205318 T^{8} - 4569485596056 p^{4} T^{10} + 450221020 p^{8} T^{12} - 29352 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 83 | \( ( 1 + 314 T + 49298 T^{2} + 5976238 T^{3} + 710509436 T^{4} + 80065592170 T^{5} + 7971612081774 T^{6} + 720873180389262 T^{7} + 61472843238255430 T^{8} + 720873180389262 p^{2} T^{9} + 7971612081774 p^{4} T^{10} + 80065592170 p^{6} T^{11} + 710509436 p^{8} T^{12} + 5976238 p^{10} T^{13} + 49298 p^{12} T^{14} + 314 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 34904 T^{2} + 653297788 T^{4} - 8297709387752 T^{6} + 76312672295494918 T^{8} - 8297709387752 p^{4} T^{10} + 653297788 p^{8} T^{12} - 34904 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 97 | \( ( 1 - 204 T + 20808 T^{2} - 974340 T^{3} + 173175628 T^{4} - 38751372396 T^{5} + 4776510719160 T^{6} - 285119654305476 T^{7} + 15660829534262118 T^{8} - 285119654305476 p^{2} T^{9} + 4776510719160 p^{4} T^{10} - 38751372396 p^{6} T^{11} + 173175628 p^{8} T^{12} - 974340 p^{10} T^{13} + 20808 p^{12} T^{14} - 204 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.91740343261835188517283199319, −2.80997946343360883902956655104, −2.76230203975103849126674481798, −2.71350076415864749629945095657, −2.68373363283124944215182934787, −2.67584926959568343051486646294, −2.48494722681661386326733246547, −2.46636847785123429347282532603, −2.25914140051998445262331148740, −2.08782809016943893630281827246, −1.94433075962368101627330774123, −1.77216387599932805321099151144, −1.70461386705622900181066435431, −1.63806818246601852560641280000, −1.56037140584904447081736509048, −1.42202092744251926264966149522, −1.40566357322861870950805958536, −1.35156125308957873218789195846, −1.13646787595854651668291232237, −1.05729380075388805768938987867, −0.914075934708151832201868139359, −0.46429444572302942533745884544, −0.31010213087057619104981208800, −0.30262613290858800484371311277, −0.27622178640926530399001641263,
0.27622178640926530399001641263, 0.30262613290858800484371311277, 0.31010213087057619104981208800, 0.46429444572302942533745884544, 0.914075934708151832201868139359, 1.05729380075388805768938987867, 1.13646787595854651668291232237, 1.35156125308957873218789195846, 1.40566357322861870950805958536, 1.42202092744251926264966149522, 1.56037140584904447081736509048, 1.63806818246601852560641280000, 1.70461386705622900181066435431, 1.77216387599932805321099151144, 1.94433075962368101627330774123, 2.08782809016943893630281827246, 2.25914140051998445262331148740, 2.46636847785123429347282532603, 2.48494722681661386326733246547, 2.67584926959568343051486646294, 2.68373363283124944215182934787, 2.71350076415864749629945095657, 2.76230203975103849126674481798, 2.80997946343360883902956655104, 2.91740343261835188517283199319
Plot not available for L-functions of degree greater than 10.