| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·9-s + 2·11-s − 2·13-s − 14-s + 16-s − 6·17-s + 3·18-s − 2·22-s − 5·25-s + 2·26-s + 28-s − 4·29-s − 32-s + 6·34-s − 3·36-s + 2·37-s + 41-s + 2·44-s − 8·47-s + 49-s + 5·50-s − 2·52-s + 4·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s + 0.603·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.426·22-s − 25-s + 0.392·26-s + 0.188·28-s − 0.742·29-s − 0.176·32-s + 1.02·34-s − 1/2·36-s + 0.328·37-s + 0.156·41-s + 0.301·44-s − 1.16·47-s + 1/7·49-s + 0.707·50-s − 0.277·52-s + 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3198160719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3198160719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.01900062362374, −12.43011196956407, −11.96145144572657, −11.35166654436864, −11.29528867265297, −10.84158007242895, −10.08909155468530, −9.687343039200850, −9.207962141335738, −8.810695196770326, −8.315792743362473, −7.943345838193859, −7.310077337397437, −6.904033280388250, −6.255815790882338, −5.967253620599449, −5.254348289879888, −4.794180453358716, −4.080296929167067, −3.628322699072075, −2.846605450597078, −2.296894232281066, −1.889446161708608, −1.123409714358768, −0.1837528837620510,
0.1837528837620510, 1.123409714358768, 1.889446161708608, 2.296894232281066, 2.846605450597078, 3.628322699072075, 4.080296929167067, 4.794180453358716, 5.254348289879888, 5.967253620599449, 6.255815790882338, 6.904033280388250, 7.310077337397437, 7.943345838193859, 8.315792743362473, 8.810695196770326, 9.207962141335738, 9.687343039200850, 10.08909155468530, 10.84158007242895, 11.29528867265297, 11.35166654436864, 11.96145144572657, 12.43011196956407, 13.01900062362374
