| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s − 2·9-s + 10-s − 2·11-s + 12-s − 3·13-s − 14-s + 15-s + 16-s − 2·18-s + 19-s + 20-s − 21-s − 2·22-s + 9·23-s + 24-s + 25-s − 3·26-s − 5·27-s − 28-s − 5·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.471·18-s + 0.229·19-s + 0.223·20-s − 0.218·21-s − 0.426·22-s + 1.87·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s − 0.962·27-s − 0.188·28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 617 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.85617338237303, −14.64285451471503, −13.76415727962502, −13.38698975204937, −13.11495492763285, −12.54910270386805, −11.86630900374318, −11.38058396770365, −10.89259945052282, −10.19471446588803, −9.716664954446516, −9.157389127982931, −8.694833820821847, −7.803986156003314, −7.631082247019274, −6.732512744785601, −6.376225719958485, −5.564372856450800, −5.124193406666046, −4.668713294023203, −3.708138753252659, −3.099595716666390, −2.686868697621105, −2.137018356273826, −1.142543546697594, 0,
1.142543546697594, 2.137018356273826, 2.686868697621105, 3.099595716666390, 3.708138753252659, 4.668713294023203, 5.124193406666046, 5.564372856450800, 6.376225719958485, 6.732512744785601, 7.631082247019274, 7.803986156003314, 8.694833820821847, 9.157389127982931, 9.716664954446516, 10.19471446588803, 10.89259945052282, 11.38058396770365, 11.86630900374318, 12.54910270386805, 13.11495492763285, 13.38698975204937, 13.76415727962502, 14.64285451471503, 14.85617338237303
