| L(s) = 1 | − 4.91·2-s + 243·3-s − 2.02e3·4-s − 1.11e3·5-s − 1.19e3·6-s − 2.14e4·7-s + 2.00e4·8-s + 5.90e4·9-s + 5.49e3·10-s + 1.10e5·11-s − 4.91e5·12-s − 3.43e4·13-s + 1.05e5·14-s − 2.71e5·15-s + 4.04e6·16-s − 8.45e6·17-s − 2.90e5·18-s + 1.50e7·19-s + 2.26e6·20-s − 5.20e6·21-s − 5.41e5·22-s + 7.31e6·23-s + 4.86e6·24-s − 4.75e7·25-s + 1.68e5·26-s + 1.43e7·27-s + 4.33e7·28-s + ⋯ |
| L(s) = 1 | − 0.108·2-s + 0.577·3-s − 0.988·4-s − 0.159·5-s − 0.0627·6-s − 0.481·7-s + 0.215·8-s + 0.333·9-s + 0.0173·10-s + 0.206·11-s − 0.570·12-s − 0.0256·13-s + 0.0523·14-s − 0.0923·15-s + 0.964·16-s − 1.44·17-s − 0.0362·18-s + 1.39·19-s + 0.158·20-s − 0.278·21-s − 0.0224·22-s + 0.236·23-s + 0.124·24-s − 0.974·25-s + 0.00278·26-s + 0.192·27-s + 0.475·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 + 7.14e8T \) |
| good | 2 | \( 1 + 4.91T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.11e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 2.14e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.10e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 3.43e4T + 1.79e12T^{2} \) |
| 17 | \( 1 + 8.45e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.50e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 7.31e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 3.39e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.38e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.04e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 6.49e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.25e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.92e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.25e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 8.19e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 8.86e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 3.46e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 5.04e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.53e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.79e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 9.94e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.07e11T + 7.15e21T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.738380786014053487474576494118, −9.297791313847486020905854936268, −8.269059949410790864833666976718, −7.32552823642821139223271006192, −5.99173983581204080049109437373, −4.65683233204665785751561380377, −3.80071989601146801537384444886, −2.65405460579938918716553734062, −1.14372295037178094476659407697, 0,
1.14372295037178094476659407697, 2.65405460579938918716553734062, 3.80071989601146801537384444886, 4.65683233204665785751561380377, 5.99173983581204080049109437373, 7.32552823642821139223271006192, 8.269059949410790864833666976718, 9.297791313847486020905854936268, 9.738380786014053487474576494118
