| L(s) = 1 | − 2-s − 3.34·3-s + 4-s + 1.93·5-s + 3.34·6-s − 8-s + 8.19·9-s − 1.93·10-s − 6.46·11-s − 3.34·12-s + 1.55·13-s − 6.46·15-s + 16-s + 5.65·17-s − 8.19·18-s + 0.378·19-s + 1.93·20-s + 6.46·22-s + 2·23-s + 3.34·24-s − 1.26·25-s − 1.55·26-s − 17.3·27-s − 29-s + 6.46·30-s − 4.00·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.93·3-s + 0.5·4-s + 0.863·5-s + 1.36·6-s − 0.353·8-s + 2.73·9-s − 0.610·10-s − 1.94·11-s − 0.965·12-s + 0.430·13-s − 1.66·15-s + 0.250·16-s + 1.37·17-s − 1.93·18-s + 0.0869·19-s + 0.431·20-s + 1.37·22-s + 0.417·23-s + 0.683·24-s − 0.253·25-s − 0.304·26-s − 3.34·27-s − 0.185·29-s + 1.18·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 3.34T + 3T^{2} \) |
| 5 | \( 1 - 1.93T + 5T^{2} \) |
| 11 | \( 1 + 6.46T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 - 0.378T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 31 | \( 1 + 4.00T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 - 2.07T + 41T^{2} \) |
| 43 | \( 1 + 0.267T + 43T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 + 7.53T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4.53T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 4.14T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 6.96T + 97T^{2} \) |
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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
−8.268507801127575369524519857551, −7.53547984252254276718718607860, −6.83290587034305677385892126942, −5.99509346719385495076084568756, −5.31804811176987880050623746339, −5.16071609587773319687725882551, −3.59445183369259526530429553371, −2.18808283983612251085083565671, −1.16211424586762208055039876015, 0,
1.16211424586762208055039876015, 2.18808283983612251085083565671, 3.59445183369259526530429553371, 5.16071609587773319687725882551, 5.31804811176987880050623746339, 5.99509346719385495076084568756, 6.83290587034305677385892126942, 7.53547984252254276718718607860, 8.268507801127575369524519857551
