| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s + 6·11-s − 12-s + 13-s − 2·14-s + 15-s + 16-s − 6·17-s − 18-s − 6·19-s − 20-s − 2·21-s − 6·22-s − 4·23-s + 24-s + 25-s − 26-s − 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.436·21-s − 1.27·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.69221600235828, −12.02792739325328, −11.55476853882151, −11.46960299458358, −10.82010505231570, −10.78042747764732, −9.968740715592797, −9.431277812393176, −9.122224421406188, −8.622316032393879, −8.167272777873697, −7.862527046345107, −7.037027074942690, −6.778287172213640, −6.350707659582101, −5.975942743751997, −5.281064221682612, −4.607764374637510, −4.105608355334884, −3.989645119486454, −3.207594134871169, −2.234891580080860, −1.893043414709551, −1.407065671359730, −0.6501214191003047, 0,
0.6501214191003047, 1.407065671359730, 1.893043414709551, 2.234891580080860, 3.207594134871169, 3.989645119486454, 4.105608355334884, 4.607764374637510, 5.281064221682612, 5.975942743751997, 6.350707659582101, 6.778287172213640, 7.037027074942690, 7.862527046345107, 8.167272777873697, 8.622316032393879, 9.122224421406188, 9.431277812393176, 9.968740715592797, 10.78042747764732, 10.82010505231570, 11.46960299458358, 11.55476853882151, 12.02792739325328, 12.69221600235828
