| L(s) = 1 | − 0.792·2-s − 1.37·4-s + 1.55·5-s − 2.36·7-s + 2.67·8-s − 1.23·10-s + 4.65·11-s + 0.117·13-s + 1.86·14-s + 0.629·16-s − 0.0922·17-s + 19-s − 2.13·20-s − 3.68·22-s + 3.26·23-s − 2.57·25-s − 0.0931·26-s + 3.24·28-s − 1.10·29-s − 1.15·31-s − 5.84·32-s + 0.0730·34-s − 3.67·35-s − 1.90·37-s − 0.792·38-s + 4.16·40-s − 9.66·41-s + ⋯ |
| L(s) = 1 | − 0.560·2-s − 0.686·4-s + 0.696·5-s − 0.892·7-s + 0.944·8-s − 0.390·10-s + 1.40·11-s + 0.0326·13-s + 0.499·14-s + 0.157·16-s − 0.0223·17-s + 0.229·19-s − 0.478·20-s − 0.785·22-s + 0.680·23-s − 0.514·25-s − 0.0182·26-s + 0.612·28-s − 0.206·29-s − 0.207·31-s − 1.03·32-s + 0.0125·34-s − 0.621·35-s − 0.312·37-s − 0.128·38-s + 0.657·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 0.792T + 2T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 + 2.36T + 7T^{2} \) |
| 11 | \( 1 - 4.65T + 11T^{2} \) |
| 13 | \( 1 - 0.117T + 13T^{2} \) |
| 17 | \( 1 + 0.0922T + 17T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 + 1.10T + 29T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 53 | \( 1 + 3.18T + 53T^{2} \) |
| 59 | \( 1 - 4.00T + 59T^{2} \) |
| 61 | \( 1 - 9.66T + 61T^{2} \) |
| 67 | \( 1 - 0.363T + 67T^{2} \) |
| 71 | \( 1 + 5.89T + 71T^{2} \) |
| 73 | \( 1 + 5.55T + 73T^{2} \) |
| 79 | \( 1 + 4.42T + 79T^{2} \) |
| 83 | \( 1 - 4.28T + 83T^{2} \) |
| 89 | \( 1 + 5.48T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.45292060623284482046649135322, −6.77966812399534619458099473311, −6.24890016232612262077962474868, −5.40493536681495771300580793268, −4.71731936096117174642718003859, −3.75702083151138944732161003243, −3.28124384204150490344229285504, −1.93352826680822190559836782873, −1.20442825083515871973406101834, 0,
1.20442825083515871973406101834, 1.93352826680822190559836782873, 3.28124384204150490344229285504, 3.75702083151138944732161003243, 4.71731936096117174642718003859, 5.40493536681495771300580793268, 6.24890016232612262077962474868, 6.77966812399534619458099473311, 7.45292060623284482046649135322
