| L(s) = 1 | + 0.384·2-s + 3-s − 1.85·4-s − 1.18·5-s + 0.384·6-s + 2.30·7-s − 1.48·8-s + 9-s − 0.454·10-s − 5.84·11-s − 1.85·12-s + 5.63·13-s + 0.887·14-s − 1.18·15-s + 3.13·16-s − 17-s + 0.384·18-s − 6.88·19-s + 2.18·20-s + 2.30·21-s − 2.24·22-s + 8.27·23-s − 1.48·24-s − 3.60·25-s + 2.16·26-s + 27-s − 4.27·28-s + ⋯ |
| L(s) = 1 | + 0.271·2-s + 0.577·3-s − 0.926·4-s − 0.528·5-s + 0.156·6-s + 0.872·7-s − 0.523·8-s + 0.333·9-s − 0.143·10-s − 1.76·11-s − 0.534·12-s + 1.56·13-s + 0.237·14-s − 0.304·15-s + 0.783·16-s − 0.242·17-s + 0.0906·18-s − 1.58·19-s + 0.489·20-s + 0.503·21-s − 0.478·22-s + 1.72·23-s − 0.302·24-s − 0.720·25-s + 0.424·26-s + 0.192·27-s − 0.807·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 0.384T + 2T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 + 5.84T + 11T^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 19 | \( 1 + 6.88T + 19T^{2} \) |
| 23 | \( 1 - 8.27T + 23T^{2} \) |
| 29 | \( 1 + 5.09T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 - 1.13T + 37T^{2} \) |
| 41 | \( 1 + 5.24T + 41T^{2} \) |
| 43 | \( 1 - 7.14T + 43T^{2} \) |
| 47 | \( 1 + 9.10T + 47T^{2} \) |
| 53 | \( 1 + 3.67T + 53T^{2} \) |
| 59 | \( 1 + 5.41T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 2.99T + 71T^{2} \) |
| 73 | \( 1 + 4.69T + 73T^{2} \) |
| 83 | \( 1 - 1.18T + 83T^{2} \) |
| 89 | \( 1 + 3.86T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.130042342748165370830225667175, −7.77068840255075028044854203401, −6.57383454361691928271381074337, −5.67275866709596238022593372912, −4.85264095055426823081141411358, −4.34669243454357351293869518550, −3.49372085000276929259679774599, −2.67247835329989919081780897475, −1.43196449881181863349287141504, 0,
1.43196449881181863349287141504, 2.67247835329989919081780897475, 3.49372085000276929259679774599, 4.34669243454357351293869518550, 4.85264095055426823081141411358, 5.67275866709596238022593372912, 6.57383454361691928271381074337, 7.77068840255075028044854203401, 8.130042342748165370830225667175
