| L(s) = 1 | + 2-s + 2·3-s − 4-s + 2·6-s + 2·7-s − 3·8-s + 9-s − 11-s − 2·12-s − 6·13-s + 2·14-s − 16-s + 18-s − 19-s + 4·21-s − 22-s − 6·24-s − 6·26-s − 4·27-s − 2·28-s + 6·29-s + 4·31-s + 5·32-s − 2·33-s − 36-s − 38-s − 12·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 1.66·13-s + 0.534·14-s − 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.872·21-s − 0.213·22-s − 1.22·24-s − 1.17·26-s − 0.769·27-s − 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.348·33-s − 1/6·36-s − 0.162·38-s − 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.87941889728263077745852361747, −7.38682238393382623015550590136, −6.28949434640367691127211952413, −5.45659190590778980854173219149, −4.56235485660288802726497695488, −4.42641737224181708589660570239, −3.01650920737381523207535582378, −2.83404334495183562463542515040, −1.69291364109964870454899368966, 0,
1.69291364109964870454899368966, 2.83404334495183562463542515040, 3.01650920737381523207535582378, 4.42641737224181708589660570239, 4.56235485660288802726497695488, 5.45659190590778980854173219149, 6.28949434640367691127211952413, 7.38682238393382623015550590136, 7.87941889728263077745852361747
