| L(s) = 1 | + 1.47·3-s − 5-s − 1.11·7-s − 0.830·9-s + 2.22·11-s + 1.47·13-s − 1.47·15-s − 4.06·17-s − 5.51·19-s − 1.64·21-s − 1.24·23-s + 25-s − 5.64·27-s − 29-s − 1.83·31-s + 3.28·33-s + 1.11·35-s + 1.05·37-s + 2.16·39-s − 4.22·41-s − 7.83·43-s + 0.830·45-s − 2.71·47-s − 5.75·49-s − 5.98·51-s + 9.34·53-s − 2.22·55-s + ⋯ |
| L(s) = 1 | + 0.850·3-s − 0.447·5-s − 0.421·7-s − 0.276·9-s + 0.672·11-s + 0.408·13-s − 0.380·15-s − 0.984·17-s − 1.26·19-s − 0.358·21-s − 0.259·23-s + 0.200·25-s − 1.08·27-s − 0.185·29-s − 0.328·31-s + 0.571·33-s + 0.188·35-s + 0.173·37-s + 0.347·39-s − 0.660·41-s − 1.19·43-s + 0.123·45-s − 0.396·47-s − 0.822·49-s − 0.837·51-s + 1.28·53-s − 0.300·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.47T + 3T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 + 5.51T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 + 7.83T + 43T^{2} \) |
| 47 | \( 1 + 2.71T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 + 0.904T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 1.43T + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 8.98T + 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.476115930489581886585101313298, −8.216016747727949413247994822561, −6.97220408246874266666177501907, −6.50552074485786732354253007220, −5.48752470563740399073650462925, −4.26090051952217487449287753366, −3.71313795257993192712601048438, −2.76602978150688915331095111334, −1.78671930937933840668810394386, 0,
1.78671930937933840668810394386, 2.76602978150688915331095111334, 3.71313795257993192712601048438, 4.26090051952217487449287753366, 5.48752470563740399073650462925, 6.50552074485786732354253007220, 6.97220408246874266666177501907, 8.216016747727949413247994822561, 8.476115930489581886585101313298
