| L(s) = 1 | + 64·4-s − 836·7-s − 440·13-s + 1.02e3·16-s − 1.33e4·19-s − 1.16e4·25-s − 5.35e4·28-s − 1.60e5·31-s − 3.01e5·37-s − 9.01e4·43-s + 6.86e5·49-s − 2.81e4·52-s − 5.84e5·61-s − 6.55e4·64-s + 7.66e5·67-s + 3.14e6·73-s − 8.56e5·76-s − 3.23e5·79-s + 3.67e5·91-s − 4.43e6·97-s − 7.45e5·100-s + 3.36e5·103-s − 7.35e6·109-s − 8.56e5·112-s − 4.42e6·121-s − 1.02e7·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s − 2.43·7-s − 0.200·13-s + 1/4·16-s − 1.95·19-s − 0.745·25-s − 2.43·28-s − 5.37·31-s − 5.94·37-s − 1.13·43-s + 5.83·49-s − 0.200·52-s − 2.57·61-s − 1/4·64-s + 2.54·67-s + 8.09·73-s − 1.95·76-s − 0.655·79-s + 0.488·91-s − 4.85·97-s − 0.745·100-s + 0.307·103-s − 5.67·109-s − 0.609·112-s − 2.49·121-s − 5.37·124-s + 4.75·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1329630567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1329630567\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 466 p^{2} T^{2} + 8857 p^{6} T^{4} - 366051854 p^{6} T^{6} - 237396228764 p^{8} T^{8} - 366051854 p^{18} T^{10} + 8857 p^{30} T^{12} + 466 p^{38} T^{14} + p^{48} T^{16} \) |
| 7 | \( ( 1 + 418 T - 11561 p T^{2} + 8507554 T^{3} + 30960042724 T^{4} + 8507554 p^{6} T^{5} - 11561 p^{13} T^{6} + 418 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 11 | \( 1 + 4420066 T^{2} + 921961559075 p T^{4} + 13784194795883091874 T^{6} + \)\(17\!\cdots\!44\)\( T^{8} + 13784194795883091874 p^{12} T^{10} + 921961559075 p^{25} T^{12} + 4420066 p^{36} T^{14} + p^{48} T^{16} \) |
| 13 | \( ( 1 + 220 T + 1673434 T^{2} - 2481303440 T^{3} - 20890967176925 T^{4} - 2481303440 p^{6} T^{5} + 1673434 p^{12} T^{6} + 220 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 17 | \( ( 1 - 56197372 T^{2} + 1562088612242310 T^{4} - 56197372 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 19 | \( ( 1 + 176 p T + 47525298 T^{2} + 176 p^{7} T^{3} + p^{12} T^{4} )^{4} \) |
| 23 | \( 1 + 419760292 T^{2} + 95595779300714506 T^{4} + \)\(15\!\cdots\!72\)\( T^{6} + \)\(21\!\cdots\!79\)\( T^{8} + \)\(15\!\cdots\!72\)\( p^{12} T^{10} + 95595779300714506 p^{24} T^{12} + 419760292 p^{36} T^{14} + p^{48} T^{16} \) |
| 29 | \( 1 + 2090745436 T^{2} + 2578286932519529962 T^{4} + \)\(22\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!39\)\( T^{8} + \)\(22\!\cdots\!72\)\( p^{12} T^{10} + 2578286932519529962 p^{24} T^{12} + 2090745436 p^{36} T^{14} + p^{48} T^{16} \) |
| 31 | \( ( 1 + 80086 T + 3040566985 T^{2} + 127992888522214 T^{3} + 4898576055954234484 T^{4} + 127992888522214 p^{6} T^{5} + 3040566985 p^{12} T^{6} + 80086 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 37 | \( ( 1 + 75272 T + 6285807906 T^{2} + 75272 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 41 | \( 1 + 5889502372 T^{2} + 18454904349512034826 T^{4} - \)\(17\!\cdots\!88\)\( T^{6} - \)\(10\!\cdots\!01\)\( T^{8} - \)\(17\!\cdots\!88\)\( p^{12} T^{10} + 18454904349512034826 p^{24} T^{12} + 5889502372 p^{36} T^{14} + p^{48} T^{16} \) |
| 43 | \( ( 1 + 45076 T - 1141082918 T^{2} - 426860587782704 T^{3} - 42736189109727161789 T^{4} - 426860587782704 p^{6} T^{5} - 1141082918 p^{12} T^{6} + 45076 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 47 | \( 1 + 6782031292 T^{2} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(20\!\cdots\!28\)\( T^{6} - \)\(12\!\cdots\!01\)\( T^{8} - \)\(20\!\cdots\!28\)\( p^{12} T^{10} + \)\(12\!\cdots\!66\)\( p^{24} T^{12} + 6782031292 p^{36} T^{14} + p^{48} T^{16} \) |
| 53 | \( ( 1 + 28851606590 T^{2} + \)\(11\!\cdots\!19\)\( T^{4} + 28851606590 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 59 | \( 1 + 156087793948 T^{2} + \)\(14\!\cdots\!94\)\( T^{4} + \)\(95\!\cdots\!04\)\( T^{6} + \)\(46\!\cdots\!15\)\( T^{8} + \)\(95\!\cdots\!04\)\( p^{12} T^{10} + \)\(14\!\cdots\!94\)\( p^{24} T^{12} + 156087793948 p^{36} T^{14} + p^{48} T^{16} \) |
| 61 | \( ( 1 + 292072 T - 36790368002 T^{2} + 5565628367905408 T^{3} + \)\(78\!\cdots\!51\)\( T^{4} + 5565628367905408 p^{6} T^{5} - 36790368002 p^{12} T^{6} + 292072 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 67 | \( ( 1 - 383396 T - 67547469158 T^{2} - 12890999480851856 T^{3} + \)\(24\!\cdots\!11\)\( T^{4} - 12890999480851856 p^{6} T^{5} - 67547469158 p^{12} T^{6} - 383396 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 71 | \( ( 1 - 273680879548 T^{2} + \)\(49\!\cdots\!90\)\( T^{4} - 273680879548 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 73 | \( ( 1 - 10786 p T + 457047261267 T^{2} - 10786 p^{7} T^{3} + p^{12} T^{4} )^{4} \) |
| 79 | \( ( 1 + 161500 T + 115654029706 T^{2} - 92983090555802000 T^{3} - \)\(57\!\cdots\!05\)\( T^{4} - 92983090555802000 p^{6} T^{5} + 115654029706 p^{12} T^{6} + 161500 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 83 | \( 1 + 733177338322 T^{2} + \)\(19\!\cdots\!49\)\( T^{4} + \)\(95\!\cdots\!86\)\( T^{6} + \)\(49\!\cdots\!80\)\( T^{8} + \)\(95\!\cdots\!86\)\( p^{12} T^{10} + \)\(19\!\cdots\!49\)\( p^{24} T^{12} + 733177338322 p^{36} T^{14} + p^{48} T^{16} \) |
| 89 | \( ( 1 - 995915389756 T^{2} + \)\(73\!\cdots\!54\)\( T^{4} - 995915389756 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 97 | \( ( 1 + 2216470 T + 2436410810089 T^{2} + 1796192801839095910 T^{3} + \)\(13\!\cdots\!80\)\( T^{4} + 1796192801839095910 p^{6} T^{5} + 2436410810089 p^{12} T^{6} + 2216470 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.68587368786603931717451768700, −4.35168460488024741992272634696, −4.13262461719487822607315591648, −4.00379288572437994646021002131, −3.94144317848805113516100406979, −3.65886327391735437265839266736, −3.61432212798094898702316928604, −3.53248102491999935232859403863, −3.44936524781860898828457962071, −3.32493854493522359238758080689, −2.82767074129197895506484309438, −2.69861920301645846882080304269, −2.55356895308888274820864646882, −2.35693280627157580403841447769, −2.21852094005953603881247864648, −1.99000285295231285330517753250, −1.80126504033232866667128070529, −1.70686427291313746111801877580, −1.54046493613576078617673146730, −1.17155730287075459559932138939, −1.13954202397495634875074667169, −0.42357019529693825410030169697, −0.36314901612746936616025704562, −0.30882989207305171023145250711, −0.05837791845564805264263060106,
0.05837791845564805264263060106, 0.30882989207305171023145250711, 0.36314901612746936616025704562, 0.42357019529693825410030169697, 1.13954202397495634875074667169, 1.17155730287075459559932138939, 1.54046493613576078617673146730, 1.70686427291313746111801877580, 1.80126504033232866667128070529, 1.99000285295231285330517753250, 2.21852094005953603881247864648, 2.35693280627157580403841447769, 2.55356895308888274820864646882, 2.69861920301645846882080304269, 2.82767074129197895506484309438, 3.32493854493522359238758080689, 3.44936524781860898828457962071, 3.53248102491999935232859403863, 3.61432212798094898702316928604, 3.65886327391735437265839266736, 3.94144317848805113516100406979, 4.00379288572437994646021002131, 4.13262461719487822607315591648, 4.35168460488024741992272634696, 4.68587368786603931717451768700
Plot not available for L-functions of degree greater than 10.
