| L(s) = 1 | + 2.46·2-s + 4.09·4-s − 2.43·5-s + 2.33·7-s + 5.17·8-s − 6.01·10-s + 11-s + 4.71·13-s + 5.77·14-s + 4.59·16-s − 3.20·17-s + 7.77·19-s − 9.98·20-s + 2.46·22-s + 2.75·23-s + 0.935·25-s + 11.6·26-s + 9.58·28-s − 2.37·29-s − 1.37·31-s + 0.987·32-s − 7.92·34-s − 5.69·35-s − 8.47·37-s + 19.1·38-s − 12.6·40-s + 3.54·41-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 2.04·4-s − 1.08·5-s + 0.883·7-s + 1.83·8-s − 1.90·10-s + 0.301·11-s + 1.30·13-s + 1.54·14-s + 1.14·16-s − 0.778·17-s + 1.78·19-s − 2.23·20-s + 0.526·22-s + 0.575·23-s + 0.187·25-s + 2.28·26-s + 1.81·28-s − 0.440·29-s − 0.246·31-s + 0.174·32-s − 1.35·34-s − 0.963·35-s − 1.39·37-s + 3.11·38-s − 1.99·40-s + 0.553·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.157502667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.157502667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 5 | \( 1 + 2.43T + 5T^{2} \) |
| 7 | \( 1 - 2.33T + 7T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 - 7.77T + 19T^{2} \) |
| 23 | \( 1 - 2.75T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 + 1.37T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 - 0.207T + 47T^{2} \) |
| 53 | \( 1 + 9.11T + 53T^{2} \) |
| 59 | \( 1 + 0.241T + 59T^{2} \) |
| 61 | \( 1 - 1.66T + 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + 2.37T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 8.93T + 97T^{2} \) |
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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
−10.74348110629511045284841833164, −9.166998992933760034747730701383, −8.149881100611765377056232564580, −7.35451436317705957695119966963, −6.52184127345309824142064906821, −5.44989752820091883251906654476, −4.72846956073830066608982671363, −3.81486043970888463737891111458, −3.20958398365823596806459979606, −1.59029122202617954127435993087,
1.59029122202617954127435993087, 3.20958398365823596806459979606, 3.81486043970888463737891111458, 4.72846956073830066608982671363, 5.44989752820091883251906654476, 6.52184127345309824142064906821, 7.35451436317705957695119966963, 8.149881100611765377056232564580, 9.166998992933760034747730701383, 10.74348110629511045284841833164
