| L(s) = 1 | + 2-s + 1.82·3-s + 4-s + 0.255·5-s + 1.82·6-s − 4.20·7-s + 8-s + 0.348·9-s + 0.255·10-s − 0.555·11-s + 1.82·12-s − 0.400·13-s − 4.20·14-s + 0.467·15-s + 16-s + 0.412·17-s + 0.348·18-s + 2.63·19-s + 0.255·20-s − 7.69·21-s − 0.555·22-s + 23-s + 1.82·24-s − 4.93·25-s − 0.400·26-s − 4.85·27-s − 4.20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.05·3-s + 0.5·4-s + 0.114·5-s + 0.747·6-s − 1.58·7-s + 0.353·8-s + 0.116·9-s + 0.0807·10-s − 0.167·11-s + 0.528·12-s − 0.111·13-s − 1.12·14-s + 0.120·15-s + 0.250·16-s + 0.100·17-s + 0.0821·18-s + 0.604·19-s + 0.0570·20-s − 1.67·21-s − 0.118·22-s + 0.208·23-s + 0.373·24-s − 0.986·25-s − 0.0785·26-s − 0.933·27-s − 0.794·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 - 0.255T + 5T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 + 0.555T + 11T^{2} \) |
| 13 | \( 1 + 0.400T + 13T^{2} \) |
| 17 | \( 1 - 0.412T + 17T^{2} \) |
| 19 | \( 1 - 2.63T + 19T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 + 8.03T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 2.28T + 41T^{2} \) |
| 43 | \( 1 - 9.76T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 9.34T + 53T^{2} \) |
| 59 | \( 1 - 2.62T + 59T^{2} \) |
| 61 | \( 1 + 3.86T + 61T^{2} \) |
| 67 | \( 1 + 0.100T + 67T^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 + 3.92T + 73T^{2} \) |
| 79 | \( 1 + 8.71T + 79T^{2} \) |
| 83 | \( 1 - 1.37T + 83T^{2} \) |
| 89 | \( 1 + 0.410T + 89T^{2} \) |
| 97 | \( 1 - 7.53T + 97T^{2} \) |
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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
−7.56242728199225187119411269320, −7.07613830077018454907280464457, −6.20788112293949097330912308362, −5.68208824073509114199773099450, −4.75931582387262506939440557087, −3.61983000372041851234172694253, −3.35602272169267309910521827096, −2.65321361438414220649776862841, −1.73036561899018693135586952220, 0,
1.73036561899018693135586952220, 2.65321361438414220649776862841, 3.35602272169267309910521827096, 3.61983000372041851234172694253, 4.75931582387262506939440557087, 5.68208824073509114199773099450, 6.20788112293949097330912308362, 7.07613830077018454907280464457, 7.56242728199225187119411269320
