| L(s) = 1 | − 8·11-s − 8·19-s + 16·29-s − 16·31-s + 48·49-s − 24·59-s + 16·79-s + 8·81-s − 16·101-s + 16·109-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 1.83·19-s + 2.97·29-s − 2.87·31-s + 48/7·49-s − 3.12·59-s + 1.80·79-s + 8/9·81-s − 1.59·101-s + 1.53·109-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570222502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570222502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 8 T^{4} - 20 p T^{8} + 424 T^{12} + 3334 T^{16} + 424 p^{4} T^{20} - 20 p^{9} T^{24} - 8 p^{12} T^{28} + p^{16} T^{32} \) |
| 7 | \( ( 1 - 24 T^{2} + 340 T^{4} - 72 p^{2} T^{6} + 28410 T^{8} - 72 p^{4} T^{10} + 340 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 4 T + 8 T^{2} + 36 T^{3} + 92 T^{4} + 68 T^{5} + 184 T^{6} + 324 T^{7} - 1370 T^{8} + 324 p T^{9} + 184 p^{2} T^{10} + 68 p^{3} T^{11} + 92 p^{4} T^{12} + 36 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 140 T^{4} + 49734 T^{8} - 140 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 40 T^{2} + 870 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 + 4 T + 8 T^{2} - 28 T^{3} - 484 T^{4} - 220 T^{5} + 3384 T^{6} + 22692 T^{7} + 191206 T^{8} + 22692 p T^{9} + 3384 p^{2} T^{10} - 220 p^{3} T^{11} - 484 p^{4} T^{12} - 28 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 152 T^{2} + 10564 T^{4} - 442664 T^{6} + 12324922 T^{8} - 442664 p^{2} T^{10} + 10564 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 8 T + 32 T^{2} + 72 T^{3} + 380 T^{4} - 12232 T^{5} + 88288 T^{6} - 148152 T^{7} - 290138 T^{8} - 148152 p T^{9} + 88288 p^{2} T^{10} - 12232 p^{3} T^{11} + 380 p^{4} T^{12} + 72 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 + 4 T + 52 T^{2} + 292 T^{3} + 1510 T^{4} + 292 p T^{5} + 52 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 92 T^{4} + 1830438 T^{8} - 92 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 - 136 T^{2} + 9988 T^{4} - 572632 T^{6} + 26690182 T^{8} - 572632 p^{2} T^{10} + 9988 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( 1 + 4568 T^{4} + 15432388 T^{8} + 32835546440 T^{12} + 67805368306822 T^{16} + 32835546440 p^{4} T^{20} + 15432388 p^{8} T^{24} + 4568 p^{12} T^{28} + p^{16} T^{32} \) |
| 47 | \( ( 1 + 296 T^{2} + 41044 T^{4} + 3480152 T^{6} + 197533306 T^{8} + 3480152 p^{2} T^{10} + 41044 p^{4} T^{12} + 296 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( 1 + 8488 T^{4} + 28367644 T^{8} + 65683864984 T^{12} + 173166077304070 T^{16} + 65683864984 p^{4} T^{20} + 28367644 p^{8} T^{24} + 8488 p^{12} T^{28} + p^{16} T^{32} \) |
| 59 | \( ( 1 + 12 T + 72 T^{2} + 396 T^{3} + 6748 T^{4} + 83052 T^{5} + 589176 T^{6} + 3781452 T^{7} + 22178598 T^{8} + 3781452 p T^{9} + 589176 p^{2} T^{10} + 83052 p^{3} T^{11} + 6748 p^{4} T^{12} + 396 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 2162 T^{4} + p^{4} T^{8} )^{4} \) |
| 67 | \( 1 + 6296 T^{4} + 28867012 T^{8} + 143453549192 T^{12} + 987214874886406 T^{16} + 143453549192 p^{4} T^{20} + 28867012 p^{8} T^{24} + 6296 p^{12} T^{28} + p^{16} T^{32} \) |
| 71 | \( ( 1 - 312 T^{2} + 52060 T^{4} - 5897736 T^{6} + 485188230 T^{8} - 5897736 p^{2} T^{10} + 52060 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 - 128 T^{2} + 11692 T^{4} - 338816 T^{6} + 16421926 T^{8} - 338816 p^{2} T^{10} + 11692 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 4 T + 196 T^{2} - 484 T^{3} + 18118 T^{4} - 484 p T^{5} + 196 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 83 | \( 1 - 24808 T^{4} + 301625668 T^{8} - 2745693101560 T^{12} + 20938093531246342 T^{16} - 2745693101560 p^{4} T^{20} + 301625668 p^{8} T^{24} - 24808 p^{12} T^{28} + p^{16} T^{32} \) |
| 89 | \( ( 1 - 504 T^{2} + 120796 T^{4} - 18233160 T^{6} + 1917111942 T^{8} - 18233160 p^{2} T^{10} + 120796 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 + 576 T^{2} + 159724 T^{4} + 27489216 T^{6} + 3203946342 T^{8} + 27489216 p^{2} T^{10} + 159724 p^{4} T^{12} + 576 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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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
−2.45900231975523151853405501910, −2.28365178352170160549711505823, −2.20284899219634700520171004193, −2.18610935657863910907295555368, −2.06473753549833158423968378771, −2.02403006827326467510003018192, −1.96606731845064066530493811716, −1.88182067638836927125485050073, −1.82728726406816144662071575090, −1.70585808061453445286504668740, −1.65097475132529507388118406639, −1.58348159442552750560086833185, −1.40477190237938111835545520982, −1.14822872930529835725805560578, −1.13885118340022406265081448761, −1.13100064883006902634599966392, −1.12730592642447118092165094479, −1.04197632753228201300202193829, −0.874827671805461285769377738999, −0.78777018921078102387308634973, −0.58605365306667816070841513759, −0.41884583728009316824912946815, −0.25009153185942262153584084543, −0.23514989886664444665732613059, −0.11579060296151630229680379700,
0.11579060296151630229680379700, 0.23514989886664444665732613059, 0.25009153185942262153584084543, 0.41884583728009316824912946815, 0.58605365306667816070841513759, 0.78777018921078102387308634973, 0.874827671805461285769377738999, 1.04197632753228201300202193829, 1.12730592642447118092165094479, 1.13100064883006902634599966392, 1.13885118340022406265081448761, 1.14822872930529835725805560578, 1.40477190237938111835545520982, 1.58348159442552750560086833185, 1.65097475132529507388118406639, 1.70585808061453445286504668740, 1.82728726406816144662071575090, 1.88182067638836927125485050073, 1.96606731845064066530493811716, 2.02403006827326467510003018192, 2.06473753549833158423968378771, 2.18610935657863910907295555368, 2.20284899219634700520171004193, 2.28365178352170160549711505823, 2.45900231975523151853405501910
Plot not available for L-functions of degree greater than 10.
