| L(s) = 1 | − 1.65·2-s − 3-s + 0.731·4-s + 2.84·5-s + 1.65·6-s − 0.210·7-s + 2.09·8-s + 9-s − 4.70·10-s + 5.85·11-s − 0.731·12-s + 5.94·13-s + 0.347·14-s − 2.84·15-s − 4.92·16-s + 17-s − 1.65·18-s − 6.00·19-s + 2.08·20-s + 0.210·21-s − 9.68·22-s − 1.53·23-s − 2.09·24-s + 3.10·25-s − 9.82·26-s − 27-s − 0.153·28-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 0.577·3-s + 0.365·4-s + 1.27·5-s + 0.674·6-s − 0.0794·7-s + 0.741·8-s + 0.333·9-s − 1.48·10-s + 1.76·11-s − 0.211·12-s + 1.64·13-s + 0.0928·14-s − 0.735·15-s − 1.23·16-s + 0.242·17-s − 0.389·18-s − 1.37·19-s + 0.465·20-s + 0.0458·21-s − 2.06·22-s − 0.320·23-s − 0.427·24-s + 0.620·25-s − 1.92·26-s − 0.192·27-s − 0.0290·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403640027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403640027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 5 | \( 1 - 2.84T + 5T^{2} \) |
| 7 | \( 1 + 0.210T + 7T^{2} \) |
| 11 | \( 1 - 5.85T + 11T^{2} \) |
| 13 | \( 1 - 5.94T + 13T^{2} \) |
| 19 | \( 1 + 6.00T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 - 1.55T + 31T^{2} \) |
| 37 | \( 1 - 0.569T + 37T^{2} \) |
| 41 | \( 1 + 0.500T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 6.53T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 - 7.13T + 59T^{2} \) |
| 61 | \( 1 + 1.20T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−8.141381894566758939745485724440, −6.96162438697855602276663098565, −6.36663942765120086012944565397, −6.19695000440686472187106418489, −5.17552151315959635785825294462, −4.25177743550081808879445244519, −3.62722527137801231823649346700, −2.11207175290797441077452542752, −1.49349463371675999360649980528, −0.813951542490897905480497078737,
0.813951542490897905480497078737, 1.49349463371675999360649980528, 2.11207175290797441077452542752, 3.62722527137801231823649346700, 4.25177743550081808879445244519, 5.17552151315959635785825294462, 6.19695000440686472187106418489, 6.36663942765120086012944565397, 6.96162438697855602276663098565, 8.141381894566758939745485724440
