| L(s) = 1 | − 2-s + 1.15·3-s + 4-s + 0.303·5-s − 1.15·6-s + 5.16·7-s − 8-s − 1.65·9-s − 0.303·10-s − 4.94·11-s + 1.15·12-s − 0.449·13-s − 5.16·14-s + 0.350·15-s + 16-s − 1.00·17-s + 1.65·18-s − 19-s + 0.303·20-s + 5.97·21-s + 4.94·22-s + 0.609·23-s − 1.15·24-s − 4.90·25-s + 0.449·26-s − 5.39·27-s + 5.16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.668·3-s + 0.5·4-s + 0.135·5-s − 0.472·6-s + 1.95·7-s − 0.353·8-s − 0.553·9-s − 0.0958·10-s − 1.48·11-s + 0.334·12-s − 0.124·13-s − 1.38·14-s + 0.0905·15-s + 0.250·16-s − 0.244·17-s + 0.391·18-s − 0.229·19-s + 0.0677·20-s + 1.30·21-s + 1.05·22-s + 0.127·23-s − 0.236·24-s − 0.981·25-s + 0.0880·26-s − 1.03·27-s + 0.975·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 - 0.303T + 5T^{2} \) |
| 7 | \( 1 - 5.16T + 7T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 23 | \( 1 - 0.609T + 23T^{2} \) |
| 29 | \( 1 - 7.13T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 1.35T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 4.14T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 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
−7.61424706477823428849706170109, −7.34907316619694474398700273515, −5.98899276283009888016922162570, −5.41279410058745730297319734281, −4.77845806414067195676153765617, −3.84367431704887162383387716237, −2.67398821417704552825551422830, −2.24689049852445997666257395175, −1.42604287780790001081087513526, 0,
1.42604287780790001081087513526, 2.24689049852445997666257395175, 2.67398821417704552825551422830, 3.84367431704887162383387716237, 4.77845806414067195676153765617, 5.41279410058745730297319734281, 5.98899276283009888016922162570, 7.34907316619694474398700273515, 7.61424706477823428849706170109
