| L(s) = 1 | + 2.88e7·2-s + 1.23e13·3-s − 3.51e16·4-s − 1.26e19·5-s + 3.57e20·6-s + 2.79e23·7-s − 2.05e24·8-s − 2.08e25·9-s − 3.64e26·10-s + 4.43e28·11-s − 4.36e29·12-s + 1.55e30·13-s + 8.06e30·14-s − 1.56e32·15-s + 1.20e33·16-s + 8.73e32·17-s − 6.00e32·18-s + 1.64e35·19-s + 4.45e35·20-s + 3.46e36·21-s + 1.27e36·22-s + 5.17e37·23-s − 2.54e37·24-s − 1.17e38·25-s + 4.46e37·26-s − 2.42e39·27-s − 9.85e39·28-s + ⋯ |
| L(s) = 1 | + 0.151·2-s + 0.938·3-s − 0.976·4-s − 0.759·5-s + 0.142·6-s + 1.60·7-s − 0.300·8-s − 0.119·9-s − 0.115·10-s + 1.01·11-s − 0.916·12-s + 0.360·13-s + 0.244·14-s − 0.712·15-s + 0.931·16-s + 0.126·17-s − 0.0181·18-s + 1.12·19-s + 0.741·20-s + 1.51·21-s + 0.154·22-s + 1.84·23-s − 0.281·24-s − 0.423·25-s + 0.0547·26-s − 1.05·27-s − 1.57·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(56-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+55/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(28)\) |
\(\approx\) |
\(2.434168594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434168594\) |
| \(L(\frac{57}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.88e7T + 3.60e16T^{2} \) |
| 3 | \( 1 - 1.23e13T + 1.74e26T^{2} \) |
| 5 | \( 1 + 1.26e19T + 2.77e38T^{2} \) |
| 7 | \( 1 - 2.79e23T + 3.02e46T^{2} \) |
| 11 | \( 1 - 4.43e28T + 1.89e57T^{2} \) |
| 13 | \( 1 - 1.55e30T + 1.84e61T^{2} \) |
| 17 | \( 1 - 8.73e32T + 4.72e67T^{2} \) |
| 19 | \( 1 - 1.64e35T + 2.14e70T^{2} \) |
| 23 | \( 1 - 5.17e37T + 7.85e74T^{2} \) |
| 29 | \( 1 - 1.58e40T + 2.70e80T^{2} \) |
| 31 | \( 1 - 5.89e40T + 1.05e82T^{2} \) |
| 37 | \( 1 + 1.50e43T + 1.78e86T^{2} \) |
| 41 | \( 1 - 2.49e44T + 5.04e88T^{2} \) |
| 43 | \( 1 + 2.64e44T + 6.93e89T^{2} \) |
| 47 | \( 1 + 1.75e46T + 9.23e91T^{2} \) |
| 53 | \( 1 + 7.37e46T + 6.84e94T^{2} \) |
| 59 | \( 1 - 2.54e48T + 2.49e97T^{2} \) |
| 61 | \( 1 + 1.07e49T + 1.56e98T^{2} \) |
| 67 | \( 1 - 1.06e50T + 2.71e100T^{2} \) |
| 71 | \( 1 + 1.17e51T + 6.59e101T^{2} \) |
| 73 | \( 1 - 2.92e50T + 3.03e102T^{2} \) |
| 79 | \( 1 - 2.26e52T + 2.34e104T^{2} \) |
| 83 | \( 1 - 3.10e52T + 3.54e105T^{2} \) |
| 89 | \( 1 - 4.89e53T + 1.64e107T^{2} \) |
| 97 | \( 1 - 4.57e53T + 1.87e109T^{2} \) |
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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
−19.35943162974186792353821919685, −17.59332407787385497159822231607, −14.86048417439935935483518138056, −13.91279710831300133068693251366, −11.61267383697861039540824713270, −8.964199280923750619958448691525, −7.929393418680684726188677607875, −4.84686041653671905108901001592, −3.44084370216387245829347446565, −1.15076563335679149777486733227,
1.15076563335679149777486733227, 3.44084370216387245829347446565, 4.84686041653671905108901001592, 7.929393418680684726188677607875, 8.964199280923750619958448691525, 11.61267383697861039540824713270, 13.91279710831300133068693251366, 14.86048417439935935483518138056, 17.59332407787385497159822231607, 19.35943162974186792353821919685
