| L(s) = 1 | − 2.51·2-s + 0.895·3-s + 4.31·4-s − 5-s − 2.24·6-s − 0.393·7-s − 5.80·8-s − 2.19·9-s + 2.51·10-s − 11-s + 3.86·12-s + 2.40·13-s + 0.987·14-s − 0.895·15-s + 5.97·16-s + 6.11·17-s + 5.52·18-s − 19-s − 4.31·20-s − 0.351·21-s + 2.51·22-s − 5.08·23-s − 5.20·24-s + 25-s − 6.04·26-s − 4.65·27-s − 1.69·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.517·3-s + 2.15·4-s − 0.447·5-s − 0.918·6-s − 0.148·7-s − 2.05·8-s − 0.732·9-s + 0.794·10-s − 0.301·11-s + 1.11·12-s + 0.667·13-s + 0.263·14-s − 0.231·15-s + 1.49·16-s + 1.48·17-s + 1.30·18-s − 0.229·19-s − 0.964·20-s − 0.0768·21-s + 0.535·22-s − 1.05·23-s − 1.06·24-s + 0.200·25-s − 1.18·26-s − 0.895·27-s − 0.320·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 - 0.895T + 3T^{2} \) |
| 7 | \( 1 + 0.393T + 7T^{2} \) |
| 13 | \( 1 - 2.40T + 13T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 23 | \( 1 + 5.08T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 + 0.585T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 3.62T + 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 8.26T + 67T^{2} \) |
| 71 | \( 1 + 1.66T + 71T^{2} \) |
| 73 | \( 1 - 1.82T + 73T^{2} \) |
| 79 | \( 1 + 8.29T + 79T^{2} \) |
| 83 | \( 1 + 7.26T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−9.625941539263580800799737022584, −8.375772083988376549735112138126, −8.249355640309129098772486051748, −7.46934995245584582416347639442, −6.43250808547342088312512289975, −5.51425751146220425290676782519, −3.71823568929001038757700627641, −2.79551008482474742053194147296, −1.55091740143875190927306740008, 0,
1.55091740143875190927306740008, 2.79551008482474742053194147296, 3.71823568929001038757700627641, 5.51425751146220425290676782519, 6.43250808547342088312512289975, 7.46934995245584582416347639442, 8.249355640309129098772486051748, 8.375772083988376549735112138126, 9.625941539263580800799737022584
