| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s − 2·9-s − 11-s − 12-s + 4·13-s + 14-s + 16-s − 2·17-s − 2·18-s + 19-s − 21-s − 22-s − 24-s + 4·26-s + 5·27-s + 28-s + 4·31-s + 32-s + 33-s − 2·34-s − 2·36-s + 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.301·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.229·19-s − 0.218·21-s − 0.213·22-s − 0.204·24-s + 0.784·26-s + 0.962·27-s + 0.188·28-s + 0.718·31-s + 0.176·32-s + 0.174·33-s − 0.342·34-s − 1/3·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 293 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.92930109770717, −13.57043672131377, −12.96962230693765, −12.52968593426976, −12.02468042986323, −11.36449480806604, −11.27437387770707, −10.73497893657392, −10.30282246151106, −9.550663419117210, −8.948538564151555, −8.494233949439602, −7.847048095098924, −7.546703103155429, −6.571442181882935, −6.306581061302207, −5.911931294639486, −5.177432930004532, −4.900704802037777, −4.182902241035695, −3.679850004165350, −2.880091237394887, −2.527930428141934, −1.571541829336404, −0.9734760589538413, 0,
0.9734760589538413, 1.571541829336404, 2.527930428141934, 2.880091237394887, 3.679850004165350, 4.182902241035695, 4.900704802037777, 5.177432930004532, 5.911931294639486, 6.306581061302207, 6.571442181882935, 7.546703103155429, 7.847048095098924, 8.494233949439602, 8.948538564151555, 9.550663419117210, 10.30282246151106, 10.73497893657392, 11.27437387770707, 11.36449480806604, 12.02468042986323, 12.52968593426976, 12.96962230693765, 13.57043672131377, 13.92930109770717
