| L(s) = 1 | − 2.32·2-s − 2.85·3-s + 3.40·4-s − 3.59·5-s + 6.64·6-s − 1.30·7-s − 3.25·8-s + 5.16·9-s + 8.36·10-s + 1.01·11-s − 9.71·12-s + 1.77·13-s + 3.02·14-s + 10.2·15-s + 0.770·16-s + 4.28·17-s − 11.9·18-s − 19-s − 12.2·20-s + 3.71·21-s − 2.37·22-s − 2.91·23-s + 9.31·24-s + 7.94·25-s − 4.13·26-s − 6.17·27-s − 4.42·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 1.64·3-s + 1.70·4-s − 1.60·5-s + 2.71·6-s − 0.491·7-s − 1.15·8-s + 1.72·9-s + 2.64·10-s + 0.307·11-s − 2.80·12-s + 0.492·13-s + 0.808·14-s + 2.65·15-s + 0.192·16-s + 1.03·17-s − 2.82·18-s − 0.229·19-s − 2.73·20-s + 0.811·21-s − 0.505·22-s − 0.607·23-s + 1.90·24-s + 1.58·25-s − 0.810·26-s − 1.18·27-s − 0.836·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 + T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 + 3.59T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 23 | \( 1 + 2.91T + 23T^{2} \) |
| 29 | \( 1 - 0.460T + 29T^{2} \) |
| 31 | \( 1 + 7.46T + 31T^{2} \) |
| 37 | \( 1 + 1.62T + 37T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 + 7.64T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 1.48T + 61T^{2} \) |
| 67 | \( 1 + 6.96T + 67T^{2} \) |
| 71 | \( 1 - 8.86T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 9.52T + 79T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 - 6.84T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.75892427711495634090845835907, −7.11129732573161994277075273966, −6.58955539356349495828745764748, −5.86497710783500400807217763092, −4.93597123017521767762565155085, −4.02078250967470996352764011885, −3.28716066619299702460026437412, −1.65944115993475623637388125008, −0.71130124613268509588367788920, 0,
0.71130124613268509588367788920, 1.65944115993475623637388125008, 3.28716066619299702460026437412, 4.02078250967470996352764011885, 4.93597123017521767762565155085, 5.86497710783500400807217763092, 6.58955539356349495828745764748, 7.11129732573161994277075273966, 7.75892427711495634090845835907
