| L(s) = 1 | − 0.631·2-s + 1.05·3-s − 1.60·4-s + 5-s − 0.667·6-s − 7-s + 2.27·8-s − 1.88·9-s − 0.631·10-s − 4.12·11-s − 1.69·12-s + 0.155·13-s + 0.631·14-s + 1.05·15-s + 1.76·16-s + 3.66·17-s + 1.18·18-s + 4.15·19-s − 1.60·20-s − 1.05·21-s + 2.60·22-s + 4.93·23-s + 2.40·24-s + 25-s − 0.0981·26-s − 5.16·27-s + 1.60·28-s + ⋯ |
| L(s) = 1 | − 0.446·2-s + 0.610·3-s − 0.800·4-s + 0.447·5-s − 0.272·6-s − 0.377·7-s + 0.803·8-s − 0.627·9-s − 0.199·10-s − 1.24·11-s − 0.488·12-s + 0.0431·13-s + 0.168·14-s + 0.273·15-s + 0.442·16-s + 0.889·17-s + 0.279·18-s + 0.952·19-s − 0.358·20-s − 0.230·21-s + 0.555·22-s + 1.02·23-s + 0.490·24-s + 0.200·25-s − 0.0192·26-s − 0.993·27-s + 0.302·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 0.631T + 2T^{2} \) |
| 3 | \( 1 - 1.05T + 3T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 13 | \( 1 - 0.155T + 13T^{2} \) |
| 17 | \( 1 - 3.66T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 4.81T + 31T^{2} \) |
| 37 | \( 1 + 5.09T + 37T^{2} \) |
| 41 | \( 1 + 0.486T + 41T^{2} \) |
| 43 | \( 1 - 9.96T + 43T^{2} \) |
| 47 | \( 1 - 9.22T + 47T^{2} \) |
| 53 | \( 1 + 9.20T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 2.92T + 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 - 2.98T + 83T^{2} \) |
| 89 | \( 1 + 1.51T + 89T^{2} \) |
| 97 | \( 1 + 3.54T + 97T^{2} \) |
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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
−7.54615110100615239138919277332, −7.22086950993730129845099199179, −5.74132316922221122545788185200, −5.55742569489847892206301267624, −4.78723668835838386100741661942, −3.66132871539535809654732774991, −3.11929333802039509178437361179, −2.29506970429400384804892592514, −1.14072071852201506027829755593, 0,
1.14072071852201506027829755593, 2.29506970429400384804892592514, 3.11929333802039509178437361179, 3.66132871539535809654732774991, 4.78723668835838386100741661942, 5.55742569489847892206301267624, 5.74132316922221122545788185200, 7.22086950993730129845099199179, 7.54615110100615239138919277332
