| L(s) = 1 | + 6.74e5·2-s + 1.16e9·3-s − 9.43e10·4-s − 4.30e13·5-s + 7.84e14·6-s + 4.52e16·7-s − 4.34e17·8-s + 1.35e18·9-s − 2.90e19·10-s + 7.70e18·11-s − 1.09e20·12-s − 9.07e21·13-s + 3.05e22·14-s − 5.00e22·15-s − 2.41e23·16-s − 1.17e24·17-s + 9.11e23·18-s − 9.32e24·19-s + 4.06e24·20-s + 5.26e25·21-s + 5.19e24·22-s + 3.49e26·23-s − 5.05e26·24-s + 3.85e25·25-s − 6.12e27·26-s + 1.57e27·27-s − 4.27e27·28-s + ⋯ |
| L(s) = 1 | + 0.910·2-s + 0.577·3-s − 0.171·4-s − 1.01·5-s + 0.525·6-s + 1.50·7-s − 1.06·8-s + 0.333·9-s − 0.919·10-s + 0.0379·11-s − 0.0991·12-s − 1.72·13-s + 1.36·14-s − 0.583·15-s − 0.798·16-s − 1.18·17-s + 0.303·18-s − 1.08·19-s + 0.173·20-s + 0.867·21-s + 0.0345·22-s + 0.976·23-s − 0.615·24-s + 0.0211·25-s − 1.56·26-s + 0.192·27-s − 0.257·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(20)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{41}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.16e9T \) |
| good | 2 | \( 1 - 6.74e5T + 5.49e11T^{2} \) |
| 5 | \( 1 + 4.30e13T + 1.81e27T^{2} \) |
| 7 | \( 1 - 4.52e16T + 9.09e32T^{2} \) |
| 11 | \( 1 - 7.70e18T + 4.11e40T^{2} \) |
| 13 | \( 1 + 9.07e21T + 2.77e43T^{2} \) |
| 17 | \( 1 + 1.17e24T + 9.71e47T^{2} \) |
| 19 | \( 1 + 9.32e24T + 7.43e49T^{2} \) |
| 23 | \( 1 - 3.49e26T + 1.28e53T^{2} \) |
| 29 | \( 1 + 3.14e28T + 1.08e57T^{2} \) |
| 31 | \( 1 + 1.22e29T + 1.45e58T^{2} \) |
| 37 | \( 1 + 1.74e30T + 1.44e61T^{2} \) |
| 41 | \( 1 - 2.93e31T + 7.91e62T^{2} \) |
| 43 | \( 1 - 5.16e31T + 5.07e63T^{2} \) |
| 47 | \( 1 + 3.56e31T + 1.62e65T^{2} \) |
| 53 | \( 1 + 4.34e33T + 1.76e67T^{2} \) |
| 59 | \( 1 - 6.09e34T + 1.15e69T^{2} \) |
| 61 | \( 1 - 4.89e34T + 4.24e69T^{2} \) |
| 67 | \( 1 + 6.92e35T + 1.64e71T^{2} \) |
| 71 | \( 1 + 3.69e35T + 1.58e72T^{2} \) |
| 73 | \( 1 - 1.37e36T + 4.67e72T^{2} \) |
| 79 | \( 1 + 8.93e36T + 1.01e74T^{2} \) |
| 83 | \( 1 - 8.75e36T + 6.98e74T^{2} \) |
| 89 | \( 1 - 7.78e37T + 1.06e76T^{2} \) |
| 97 | \( 1 + 8.85e38T + 3.04e77T^{2} \) |
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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
−15.10870374583037702642966807509, −14.51943940930740246825377482729, −12.75508760416339216313437763544, −11.33463359276215627565462225138, −8.857342974004666658725633650798, −7.46353779752608070597841128243, −4.92784830195079787238413911521, −4.07302561864195185001626089755, −2.29099628756857071565576161535, 0,
2.29099628756857071565576161535, 4.07302561864195185001626089755, 4.92784830195079787238413911521, 7.46353779752608070597841128243, 8.857342974004666658725633650798, 11.33463359276215627565462225138, 12.75508760416339216313437763544, 14.51943940930740246825377482729, 15.10870374583037702642966807509
