| L(s) = 1 | + 3.31e5·2-s − 1.54e8·3-s + 7.55e10·4-s + 2.12e12·5-s − 5.12e13·6-s + 3.96e14·7-s + 1.36e16·8-s − 2.61e16·9-s + 7.03e17·10-s − 6.82e17·11-s − 1.16e19·12-s − 1.17e19·13-s + 1.31e20·14-s − 3.28e20·15-s + 1.93e21·16-s − 1.33e21·17-s − 8.65e21·18-s − 3.58e22·19-s + 1.60e23·20-s − 6.13e22·21-s − 2.26e23·22-s + 6.92e23·23-s − 2.11e24·24-s + 1.58e24·25-s − 3.91e24·26-s + 1.17e25·27-s + 2.99e25·28-s + ⋯ |
| L(s) = 1 | + 1.78·2-s − 0.691·3-s + 2.19·4-s + 1.24·5-s − 1.23·6-s + 0.644·7-s + 2.14·8-s − 0.521·9-s + 2.22·10-s − 0.407·11-s − 1.52·12-s − 0.378·13-s + 1.15·14-s − 0.859·15-s + 1.63·16-s − 0.390·17-s − 0.933·18-s − 1.50·19-s + 2.73·20-s − 0.445·21-s − 0.728·22-s + 1.02·23-s − 1.48·24-s + 0.546·25-s − 0.676·26-s + 1.05·27-s + 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(36-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+35/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(18)\) |
\(\approx\) |
\(4.126595657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.126595657\) |
| \(L(\frac{37}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 3.31e5T + 3.43e10T^{2} \) |
| 3 | \( 1 + 1.54e8T + 5.00e16T^{2} \) |
| 5 | \( 1 - 2.12e12T + 2.91e24T^{2} \) |
| 7 | \( 1 - 3.96e14T + 3.78e29T^{2} \) |
| 11 | \( 1 + 6.82e17T + 2.81e36T^{2} \) |
| 13 | \( 1 + 1.17e19T + 9.72e38T^{2} \) |
| 17 | \( 1 + 1.33e21T + 1.16e43T^{2} \) |
| 19 | \( 1 + 3.58e22T + 5.70e44T^{2} \) |
| 23 | \( 1 - 6.92e23T + 4.57e47T^{2} \) |
| 29 | \( 1 + 5.67e25T + 1.52e51T^{2} \) |
| 31 | \( 1 - 1.02e26T + 1.57e52T^{2} \) |
| 37 | \( 1 - 4.50e27T + 7.71e54T^{2} \) |
| 41 | \( 1 + 6.23e27T + 2.80e56T^{2} \) |
| 43 | \( 1 + 2.75e27T + 1.48e57T^{2} \) |
| 47 | \( 1 - 1.41e29T + 3.33e58T^{2} \) |
| 53 | \( 1 + 2.48e30T + 2.23e60T^{2} \) |
| 59 | \( 1 - 5.47e30T + 9.54e61T^{2} \) |
| 61 | \( 1 - 2.30e31T + 3.06e62T^{2} \) |
| 67 | \( 1 - 1.55e31T + 8.17e63T^{2} \) |
| 71 | \( 1 + 1.13e32T + 6.22e64T^{2} \) |
| 73 | \( 1 - 3.23e32T + 1.64e65T^{2} \) |
| 79 | \( 1 + 1.44e32T + 2.61e66T^{2} \) |
| 83 | \( 1 - 3.50e33T + 1.47e67T^{2} \) |
| 89 | \( 1 - 9.37e33T + 1.69e68T^{2} \) |
| 97 | \( 1 - 3.59e33T + 3.44e69T^{2} \) |
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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
−23.63515365926797657773540655489, −22.10432661503916396384672748069, −20.90277113678176266986213773335, −17.16519091384425771067062733987, −14.67016831171737410675322590588, −13.05924299969805232161932275817, −11.10643246241256940705304121290, −6.18932512378858861994327713434, −4.94822276660157313077474188750, −2.28810375041348958159043620596,
2.28810375041348958159043620596, 4.94822276660157313077474188750, 6.18932512378858861994327713434, 11.10643246241256940705304121290, 13.05924299969805232161932275817, 14.67016831171737410675322590588, 17.16519091384425771067062733987, 20.90277113678176266986213773335, 22.10432661503916396384672748069, 23.63515365926797657773540655489
