| L(s) = 1 | + 2.51·2-s + 3-s + 4.31·4-s + 3.63·5-s + 2.51·6-s + 5.80·8-s + 9-s + 9.12·10-s + 3.40·11-s + 4.31·12-s − 0.598·13-s + 3.63·15-s + 5.96·16-s + 5.17·17-s + 2.51·18-s − 1.01·19-s + 15.6·20-s + 8.54·22-s − 6.86·23-s + 5.80·24-s + 8.19·25-s − 1.50·26-s + 27-s − 0.135·29-s + 9.12·30-s − 10.2·31-s + 3.36·32-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.577·3-s + 2.15·4-s + 1.62·5-s + 1.02·6-s + 2.05·8-s + 0.333·9-s + 2.88·10-s + 1.02·11-s + 1.24·12-s − 0.166·13-s + 0.937·15-s + 1.49·16-s + 1.25·17-s + 0.592·18-s − 0.233·19-s + 3.50·20-s + 1.82·22-s − 1.43·23-s + 1.18·24-s + 1.63·25-s − 0.295·26-s + 0.192·27-s − 0.0251·29-s + 1.66·30-s − 1.83·31-s + 0.595·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.00126169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.00126169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 11 | \( 1 - 3.40T + 11T^{2} \) |
| 13 | \( 1 + 0.598T + 13T^{2} \) |
| 17 | \( 1 - 5.17T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 + 0.135T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 6.41T + 37T^{2} \) |
| 41 | \( 1 + 4.51T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 - 7.98T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 4.83T + 59T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 2.87T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 - 0.137T + 89T^{2} \) |
| 97 | \( 1 + 6.56T + 97T^{2} \) |
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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
−7.35201175758867536186122339386, −6.82191005669625076391453247868, −6.04343161009260798595561978909, −5.68270730163596452345286613971, −5.08330884450896040304230508012, −4.10474553182551195243144756182, −3.59432988163549865502115182548, −2.78593758670667617375038713739, −1.94409169394273699981011499888, −1.53399091553144217920201981908,
1.53399091553144217920201981908, 1.94409169394273699981011499888, 2.78593758670667617375038713739, 3.59432988163549865502115182548, 4.10474553182551195243144756182, 5.08330884450896040304230508012, 5.68270730163596452345286613971, 6.04343161009260798595561978909, 6.82191005669625076391453247868, 7.35201175758867536186122339386
