| L(s) = 1 | + 2.30·2-s + 3-s + 3.31·4-s + 0.431·5-s + 2.30·6-s − 4.25·7-s + 3.03·8-s + 9-s + 0.995·10-s − 1.92·11-s + 3.31·12-s − 1.04·13-s − 9.81·14-s + 0.431·15-s + 0.369·16-s + 17-s + 2.30·18-s + 2.88·19-s + 1.43·20-s − 4.25·21-s − 4.42·22-s + 1.81·23-s + 3.03·24-s − 4.81·25-s − 2.41·26-s + 27-s − 14.1·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 0.577·3-s + 1.65·4-s + 0.193·5-s + 0.941·6-s − 1.60·7-s + 1.07·8-s + 0.333·9-s + 0.314·10-s − 0.578·11-s + 0.957·12-s − 0.290·13-s − 2.62·14-s + 0.111·15-s + 0.0924·16-s + 0.242·17-s + 0.543·18-s + 0.660·19-s + 0.320·20-s − 0.928·21-s − 0.943·22-s + 0.378·23-s + 0.620·24-s − 0.962·25-s − 0.472·26-s + 0.192·27-s − 2.66·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 5 | \( 1 - 0.431T + 5T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 - 1.81T + 23T^{2} \) |
| 29 | \( 1 + 5.88T + 29T^{2} \) |
| 31 | \( 1 + 0.0657T + 31T^{2} \) |
| 37 | \( 1 - 3.11T + 37T^{2} \) |
| 41 | \( 1 + 8.77T + 41T^{2} \) |
| 43 | \( 1 + 8.05T + 43T^{2} \) |
| 47 | \( 1 + 2.45T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 6.56T + 61T^{2} \) |
| 67 | \( 1 + 1.09T + 67T^{2} \) |
| 71 | \( 1 + 6.46T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 3.61T + 83T^{2} \) |
| 89 | \( 1 + 4.39T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.28703223817417865095960973491, −6.54409858992046249743961214544, −6.05648667010269435933512778551, −5.31216410074905960344343817743, −4.68689234503024951691900649423, −3.60997888775901402490274222320, −3.35890961982618877749344461596, −2.67771892390715297118951873694, −1.80201729358380130173381229524, 0,
1.80201729358380130173381229524, 2.67771892390715297118951873694, 3.35890961982618877749344461596, 3.60997888775901402490274222320, 4.68689234503024951691900649423, 5.31216410074905960344343817743, 6.05648667010269435933512778551, 6.54409858992046249743961214544, 7.28703223817417865095960973491
