| L(s) = 1 | − 6.79e5·2-s − 3.87e8·3-s + 3.24e11·4-s − 1.35e13·5-s + 2.63e14·6-s − 4.10e15·7-s − 1.26e17·8-s + 1.50e17·9-s + 9.19e18·10-s + 1.15e19·11-s − 1.25e20·12-s + 4.10e20·13-s + 2.78e21·14-s + 5.24e21·15-s + 4.16e22·16-s − 2.52e22·17-s − 1.01e23·18-s + 8.17e23·19-s − 4.38e24·20-s + 1.59e24·21-s − 7.82e24·22-s − 7.67e24·23-s + 4.91e25·24-s + 1.10e26·25-s − 2.79e26·26-s − 5.81e25·27-s − 1.33e27·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 0.577·3-s + 2.35·4-s − 1.58·5-s + 1.05·6-s − 0.952·7-s − 2.49·8-s + 0.333·9-s + 2.90·10-s + 0.624·11-s − 1.36·12-s + 1.01·13-s + 1.74·14-s + 0.915·15-s + 2.20·16-s − 0.435·17-s − 0.610·18-s + 1.80·19-s − 3.74·20-s + 0.550·21-s − 1.14·22-s − 0.493·23-s + 1.43·24-s + 1.51·25-s − 1.85·26-s − 0.192·27-s − 2.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3.87e8T \) |
| good | 2 | \( 1 + 6.79e5T + 1.37e11T^{2} \) |
| 5 | \( 1 + 1.35e13T + 7.27e25T^{2} \) |
| 7 | \( 1 + 4.10e15T + 1.85e31T^{2} \) |
| 11 | \( 1 - 1.15e19T + 3.40e38T^{2} \) |
| 13 | \( 1 - 4.10e20T + 1.64e41T^{2} \) |
| 17 | \( 1 + 2.52e22T + 3.36e45T^{2} \) |
| 19 | \( 1 - 8.17e23T + 2.06e47T^{2} \) |
| 23 | \( 1 + 7.67e24T + 2.42e50T^{2} \) |
| 29 | \( 1 - 7.52e26T + 1.28e54T^{2} \) |
| 31 | \( 1 + 3.41e27T + 1.51e55T^{2} \) |
| 37 | \( 1 + 4.82e28T + 1.05e58T^{2} \) |
| 41 | \( 1 - 5.14e29T + 4.70e59T^{2} \) |
| 43 | \( 1 - 1.56e30T + 2.74e60T^{2} \) |
| 47 | \( 1 + 1.18e31T + 7.37e61T^{2} \) |
| 53 | \( 1 + 4.07e31T + 6.28e63T^{2} \) |
| 59 | \( 1 - 6.63e31T + 3.32e65T^{2} \) |
| 61 | \( 1 + 1.38e33T + 1.14e66T^{2} \) |
| 67 | \( 1 - 3.53e33T + 3.67e67T^{2} \) |
| 71 | \( 1 + 6.34e33T + 3.13e68T^{2} \) |
| 73 | \( 1 - 4.38e34T + 8.76e68T^{2} \) |
| 79 | \( 1 - 2.43e34T + 1.63e70T^{2} \) |
| 83 | \( 1 - 4.41e35T + 1.01e71T^{2} \) |
| 89 | \( 1 - 1.79e36T + 1.34e72T^{2} \) |
| 97 | \( 1 + 4.46e36T + 3.24e73T^{2} \) |
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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
−16.25800865321607332381384257567, −15.81092160092555217495271714984, −12.00673509328756648229595396047, −10.99044783344975883596007816580, −9.313913502798474368289953460466, −7.79545021807776993610418432108, −6.55921782783709352127265946708, −3.46420618760276415991557687374, −1.03782586238048776179657920460, 0,
1.03782586238048776179657920460, 3.46420618760276415991557687374, 6.55921782783709352127265946708, 7.79545021807776993610418432108, 9.313913502798474368289953460466, 10.99044783344975883596007816580, 12.00673509328756648229595396047, 15.81092160092555217495271714984, 16.25800865321607332381384257567
