| L(s) = 1 | + 1.23e12·2-s + 8.35e18·3-s + 9.09e23·4-s + 3.84e27·5-s + 1.02e31·6-s + 3.79e33·7-s + 3.74e35·8-s + 2.04e37·9-s + 4.72e39·10-s + 5.69e40·11-s + 7.59e42·12-s − 1.86e44·13-s + 4.66e45·14-s + 3.20e46·15-s − 8.84e46·16-s − 4.20e48·17-s + 2.51e49·18-s + 2.63e50·19-s + 3.49e51·20-s + 3.16e52·21-s + 7.00e52·22-s + 4.17e53·23-s + 3.12e54·24-s − 1.78e54·25-s − 2.29e56·26-s − 2.40e56·27-s + 3.44e57·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.18·3-s + 1.50·4-s + 0.944·5-s + 1.88·6-s + 1.57·7-s + 0.797·8-s + 0.415·9-s + 1.49·10-s + 0.417·11-s + 1.78·12-s − 1.86·13-s + 2.49·14-s + 1.12·15-s − 0.242·16-s − 1.05·17-s + 0.657·18-s + 0.812·19-s + 1.42·20-s + 1.87·21-s + 0.659·22-s + 0.679·23-s + 0.948·24-s − 0.107·25-s − 2.95·26-s − 0.695·27-s + 2.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(80-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+79/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(40)\) |
\(\approx\) |
\(9.188453062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.188453062\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.23e12T + 6.04e23T^{2} \) |
| 3 | \( 1 - 8.35e18T + 4.92e37T^{2} \) |
| 5 | \( 1 - 3.84e27T + 1.65e55T^{2} \) |
| 7 | \( 1 - 3.79e33T + 5.79e66T^{2} \) |
| 11 | \( 1 - 5.69e40T + 1.86e82T^{2} \) |
| 13 | \( 1 + 1.86e44T + 1.00e88T^{2} \) |
| 17 | \( 1 + 4.20e48T + 1.60e97T^{2} \) |
| 19 | \( 1 - 2.63e50T + 1.05e101T^{2} \) |
| 23 | \( 1 - 4.17e53T + 3.77e107T^{2} \) |
| 29 | \( 1 + 6.41e57T + 3.38e115T^{2} \) |
| 31 | \( 1 + 3.24e57T + 6.57e117T^{2} \) |
| 37 | \( 1 - 8.16e61T + 7.72e123T^{2} \) |
| 41 | \( 1 - 6.43e63T + 2.56e127T^{2} \) |
| 43 | \( 1 - 3.95e63T + 1.10e129T^{2} \) |
| 47 | \( 1 + 4.35e65T + 1.24e132T^{2} \) |
| 53 | \( 1 - 9.71e67T + 1.65e136T^{2} \) |
| 59 | \( 1 - 6.57e68T + 7.89e139T^{2} \) |
| 61 | \( 1 - 3.50e70T + 1.09e141T^{2} \) |
| 67 | \( 1 + 4.53e71T + 1.81e144T^{2} \) |
| 71 | \( 1 + 8.10e72T + 1.77e146T^{2} \) |
| 73 | \( 1 + 5.59e73T + 1.59e147T^{2} \) |
| 79 | \( 1 - 1.00e74T + 8.17e149T^{2} \) |
| 83 | \( 1 + 3.41e75T + 4.04e151T^{2} \) |
| 89 | \( 1 - 8.47e76T + 1.00e154T^{2} \) |
| 97 | \( 1 - 2.15e78T + 9.01e156T^{2} \) |
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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
−14.83423944722484620138192932698, −14.41392983517037515006054469691, −13.27879816575469130253569812295, −11.53511110409786040199221700519, −9.256303161317202033821178661314, −7.42262459025747960756404017456, −5.44439330053608455545275736723, −4.35580474518380941313534576081, −2.63180320104243486760473805870, −1.91880601465280198807570261067,
1.91880601465280198807570261067, 2.63180320104243486760473805870, 4.35580474518380941313534576081, 5.44439330053608455545275736723, 7.42262459025747960756404017456, 9.256303161317202033821178661314, 11.53511110409786040199221700519, 13.27879816575469130253569812295, 14.41392983517037515006054469691, 14.83423944722484620138192932698
