| L(s) = 1 | + 0.239·2-s + 1.70·3-s − 1.94·4-s + 5-s + 0.407·6-s − 0.944·8-s − 0.100·9-s + 0.239·10-s + 5.19·11-s − 3.30·12-s − 6.25·13-s + 1.70·15-s + 3.65·16-s + 5.57·17-s − 0.0239·18-s − 7.11·19-s − 1.94·20-s + 1.24·22-s − 3.61·23-s − 1.60·24-s + 25-s − 1.49·26-s − 5.27·27-s + 8.27·29-s + 0.407·30-s − 31-s + 2.76·32-s + ⋯ |
| L(s) = 1 | + 0.169·2-s + 0.983·3-s − 0.971·4-s + 0.447·5-s + 0.166·6-s − 0.333·8-s − 0.0333·9-s + 0.0757·10-s + 1.56·11-s − 0.954·12-s − 1.73·13-s + 0.439·15-s + 0.914·16-s + 1.35·17-s − 0.00564·18-s − 1.63·19-s − 0.434·20-s + 0.265·22-s − 0.752·23-s − 0.328·24-s + 0.200·25-s − 0.294·26-s − 1.01·27-s + 1.53·29-s + 0.0744·30-s − 0.179·31-s + 0.488·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.239T + 2T^{2} \) |
| 3 | \( 1 - 1.70T + 3T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 - 5.57T + 17T^{2} \) |
| 19 | \( 1 + 7.11T + 19T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 37 | \( 1 - 3.38T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 + 4.22T + 53T^{2} \) |
| 59 | \( 1 + 0.482T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 3.31T + 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 4.32T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 7.50T + 97T^{2} \) |
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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
−7.87566760596143624769770735914, −6.72288303607765568067082762781, −6.24113530142019453906306129476, −5.22728394623251298285492852042, −4.65829735869691569166653653077, −3.84727247476648309729248759819, −3.22865232207241052645222397554, −2.34130241069185901599094385553, −1.43032160889501486443697894348, 0,
1.43032160889501486443697894348, 2.34130241069185901599094385553, 3.22865232207241052645222397554, 3.84727247476648309729248759819, 4.65829735869691569166653653077, 5.22728394623251298285492852042, 6.24113530142019453906306129476, 6.72288303607765568067082762781, 7.87566760596143624769770735914
