| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s − 2·11-s − 12-s + 6·13-s + 2·14-s + 16-s − 2·17-s − 18-s + 2·21-s + 2·22-s + 24-s − 6·26-s − 27-s − 2·28-s − 8·31-s − 32-s + 2·33-s + 2·34-s + 36-s − 2·37-s − 6·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 1.66·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.436·21-s + 0.426·22-s + 0.204·24-s − 1.17·26-s − 0.192·27-s − 0.377·28-s − 1.43·31-s − 0.176·32-s + 0.348·33-s + 0.342·34-s + 1/6·36-s − 0.328·37-s − 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.33224845193670, −13.54892198338057, −13.12619233747414, −12.82259565964052, −12.28814823284686, −11.51274301760342, −11.18227242111064, −10.76753320618249, −10.34437516498211, −9.698794900624901, −9.248871849547279, −8.774101259076785, −8.159168410746120, −7.717264048419326, −7.015816970616843, −6.424872808561006, −6.264150327609784, −5.459362636604442, −5.089639950646711, −4.078811969463670, −3.629919409470250, −3.030758959702200, −2.183976749989752, −1.539907717196752, −0.7359562058232900, 0,
0.7359562058232900, 1.539907717196752, 2.183976749989752, 3.030758959702200, 3.629919409470250, 4.078811969463670, 5.089639950646711, 5.459362636604442, 6.264150327609784, 6.424872808561006, 7.015816970616843, 7.717264048419326, 8.159168410746120, 8.774101259076785, 9.248871849547279, 9.698794900624901, 10.34437516498211, 10.76753320618249, 11.18227242111064, 11.51274301760342, 12.28814823284686, 12.82259565964052, 13.12619233747414, 13.54892198338057, 14.33224845193670
