| L(s) = 1 | − 2.55·2-s − 3-s + 4.51·4-s − 5-s + 2.55·6-s − 4.92·7-s − 6.41·8-s + 9-s + 2.55·10-s + 0.347·11-s − 4.51·12-s − 2.97·13-s + 12.5·14-s + 15-s + 7.34·16-s − 6.49·17-s − 2.55·18-s − 6.17·19-s − 4.51·20-s + 4.92·21-s − 0.886·22-s + 6.41·24-s + 25-s + 7.60·26-s − 27-s − 22.2·28-s − 4.05·29-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.577·3-s + 2.25·4-s − 0.447·5-s + 1.04·6-s − 1.86·7-s − 2.26·8-s + 0.333·9-s + 0.807·10-s + 0.104·11-s − 1.30·12-s − 0.825·13-s + 3.36·14-s + 0.258·15-s + 1.83·16-s − 1.57·17-s − 0.601·18-s − 1.41·19-s − 1.00·20-s + 1.07·21-s − 0.188·22-s + 1.30·24-s + 0.200·25-s + 1.49·26-s − 0.192·27-s − 4.20·28-s − 0.752·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 7 | \( 1 + 4.92T + 7T^{2} \) |
| 11 | \( 1 - 0.347T + 11T^{2} \) |
| 13 | \( 1 + 2.97T + 13T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 19 | \( 1 + 6.17T + 19T^{2} \) |
| 29 | \( 1 + 4.05T + 29T^{2} \) |
| 31 | \( 1 - 7.03T + 31T^{2} \) |
| 37 | \( 1 - 2.11T + 37T^{2} \) |
| 41 | \( 1 + 3.94T + 41T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 - 5.11T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 7.32T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 0.401T + 71T^{2} \) |
| 73 | \( 1 + 6.69T + 73T^{2} \) |
| 79 | \( 1 + 8.07T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.31319905940717337953160418238, −7.01255819677695834569676797441, −6.41184978220966629180817122366, −5.94472714589388356526566610277, −4.61123015373116197209100438901, −3.77718685022973755025553963745, −2.67954456031209867278206242361, −2.12866809325861094920520340385, −0.63176891889671616654179755745, 0,
0.63176891889671616654179755745, 2.12866809325861094920520340385, 2.67954456031209867278206242361, 3.77718685022973755025553963745, 4.61123015373116197209100438901, 5.94472714589388356526566610277, 6.41184978220966629180817122366, 7.01255819677695834569676797441, 7.31319905940717337953160418238
