| L(s) = 1 | + 4·2-s + 8·4-s + 12·5-s − 4·7-s + 8·8-s + 6·9-s + 48·10-s + 12·13-s − 16·14-s − 4·16-s + 24·18-s − 52·19-s + 96·20-s + 72·25-s + 48·26-s − 32·28-s + 72·29-s + 4·31-s − 32·32-s − 48·35-s + 48·36-s − 68·37-s − 208·38-s + 96·40-s + 60·41-s + 72·45-s − 144·47-s + ⋯ |
| L(s) = 1 | + 2·2-s + 2·4-s + 12/5·5-s − 4/7·7-s + 8-s + 2/3·9-s + 24/5·10-s + 0.923·13-s − 8/7·14-s − 1/4·16-s + 4/3·18-s − 2.73·19-s + 24/5·20-s + 2.87·25-s + 1.84·26-s − 8/7·28-s + 2.48·29-s + 4/31·31-s − 32-s − 1.37·35-s + 4/3·36-s − 1.83·37-s − 5.47·38-s + 12/5·40-s + 1.46·41-s + 8/5·45-s − 3.06·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.834102780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.834102780\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 12 T + 14 p T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2594 T^{4} - 444 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 180 T^{3} + 4034 T^{4} + 180 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 26414 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 268 T^{2} + 12198 T^{4} - 268 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 52 T + 1352 T^{2} + 36036 T^{3} + 850274 T^{4} + 36036 p^{2} T^{5} + 1352 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1228 T^{2} + 763878 T^{4} - 1228 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 36 T + 1814 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 1548 T^{3} - 1517566 T^{4} + 1548 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 68 T + 2312 T^{2} + 106284 T^{3} + 4848302 T^{4} + 106284 p^{2} T^{5} + 2312 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 60 T + 1800 T^{2} - 23820 T^{3} - 1333438 T^{4} - 23820 p^{2} T^{5} + 1800 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5380 T^{2} + 13933734 T^{4} - 5380 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 144 T + 10368 T^{2} + 14544 p T^{3} + 17486 p^{2} T^{4} + 14544 p^{3} T^{5} + 10368 p^{4} T^{6} + 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 60 T + 5750 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} - 91128 T^{3} - 24134866 T^{4} - 91128 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 96 T + 6674 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 28 T + 392 T^{2} - 33012 T^{3} - 29346142 T^{4} - 33012 p^{2} T^{5} + 392 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 70728 T^{3} + 12984782 T^{4} - 70728 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 92 T + 4232 T^{2} - 532404 T^{3} + 66768974 T^{4} - 532404 p^{2} T^{5} + 4232 p^{4} T^{6} - 92 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 144 T + 11858 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} - 670824 T^{3} - 89021458 T^{4} - 670824 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 83100 T^{3} + 94919234 T^{4} - 83100 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 164 T + 13448 T^{2} + 1429260 T^{3} + 151420814 T^{4} + 1429260 p^{2} T^{5} + 13448 p^{4} T^{6} + 164 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.58269284567613699185558686928, −10.08645766468685540650190577675, −9.854517143384924155962377121949, −9.797433138008203317680747890569, −9.603373426463757757849281407219, −8.785679004485147129329902668562, −8.576448612925182911060043559716, −8.421586010150319416309272017544, −8.412123354644109433154813164313, −7.36612495223565123544504107311, −6.94724977497096726507135375243, −6.75734404286969207946185556152, −6.36675238337253509913894055915, −6.33178979939309660747212104869, −5.99592639545284492878394082543, −5.47242680179009150713002915964, −5.32042872923809144637685446922, −4.86881179575685435150385921268, −4.42686904735446228657874972521, −3.94445375271168561737640816706, −3.83585695595220724030054914464, −2.79912030090785898018306733881, −2.72332490700667382595560767124, −2.04895953306349928450982442824, −1.45783827172181421360418103857,
1.45783827172181421360418103857, 2.04895953306349928450982442824, 2.72332490700667382595560767124, 2.79912030090785898018306733881, 3.83585695595220724030054914464, 3.94445375271168561737640816706, 4.42686904735446228657874972521, 4.86881179575685435150385921268, 5.32042872923809144637685446922, 5.47242680179009150713002915964, 5.99592639545284492878394082543, 6.33178979939309660747212104869, 6.36675238337253509913894055915, 6.75734404286969207946185556152, 6.94724977497096726507135375243, 7.36612495223565123544504107311, 8.412123354644109433154813164313, 8.421586010150319416309272017544, 8.576448612925182911060043559716, 8.785679004485147129329902668562, 9.603373426463757757849281407219, 9.797433138008203317680747890569, 9.854517143384924155962377121949, 10.08645766468685540650190577675, 10.58269284567613699185558686928
