| L(s) = 1 | + 1.60·2-s + 3-s + 0.584·4-s + 5-s + 1.60·6-s + 0.859·7-s − 2.27·8-s + 9-s + 1.60·10-s − 4.75·11-s + 0.584·12-s + 0.281·13-s + 1.38·14-s + 15-s − 4.82·16-s − 1.54·17-s + 1.60·18-s − 1.69·19-s + 0.584·20-s + 0.859·21-s − 7.64·22-s + 0.956·23-s − 2.27·24-s + 25-s + 0.451·26-s + 27-s + 0.502·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 0.577·3-s + 0.292·4-s + 0.447·5-s + 0.656·6-s + 0.324·7-s − 0.804·8-s + 0.333·9-s + 0.508·10-s − 1.43·11-s + 0.168·12-s + 0.0779·13-s + 0.369·14-s + 0.258·15-s − 1.20·16-s − 0.375·17-s + 0.378·18-s − 0.388·19-s + 0.130·20-s + 0.187·21-s − 1.62·22-s + 0.199·23-s − 0.464·24-s + 0.200·25-s + 0.0886·26-s + 0.192·27-s + 0.0949·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.60T + 2T^{2} \) |
| 7 | \( 1 - 0.859T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 - 0.281T + 13T^{2} \) |
| 17 | \( 1 + 1.54T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 - 0.956T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 + 2.13T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 - 0.653T + 41T^{2} \) |
| 43 | \( 1 - 4.50T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 - 3.80T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 + 9.72T + 61T^{2} \) |
| 67 | \( 1 - 4.08T + 67T^{2} \) |
| 71 | \( 1 - 3.22T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 7.41T + 97T^{2} \) |
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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
−7.70022667324049406984376797947, −6.93791283542632325226782808913, −6.05449304736044458572186576860, −5.41114466568931780782977747425, −4.85530549347045327645856708471, −4.09982269131355916044989680572, −3.25801561067647413347585409778, −2.56288342126169163745211812662, −1.78274982485260192356803210196, 0,
1.78274982485260192356803210196, 2.56288342126169163745211812662, 3.25801561067647413347585409778, 4.09982269131355916044989680572, 4.85530549347045327645856708471, 5.41114466568931780782977747425, 6.05449304736044458572186576860, 6.93791283542632325226782808913, 7.70022667324049406984376797947
