| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 11·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s − 22·10-s + 11·11-s − 12·12-s − 37·13-s + 14·14-s − 33·15-s + 16·16-s − 46·17-s − 18·18-s + 15·19-s + 44·20-s + 21·21-s − 22·22-s − 92·23-s + 24·24-s − 4·25-s + 74·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.983·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.695·10-s + 0.301·11-s − 0.288·12-s − 0.789·13-s + 0.267·14-s − 0.568·15-s + 1/4·16-s − 0.656·17-s − 0.235·18-s + 0.181·19-s + 0.491·20-s + 0.218·21-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 0.0319·25-s + 0.558·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 - 11 T + p^{3} T^{2} \) |
| 13 | \( 1 + 37 T + p^{3} T^{2} \) |
| 17 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 15 T + p^{3} T^{2} \) |
| 23 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 205 T + p^{3} T^{2} \) |
| 31 | \( 1 - 142 T + p^{3} T^{2} \) |
| 37 | \( 1 + 431 T + p^{3} T^{2} \) |
| 41 | \( 1 + 8 T + p^{3} T^{2} \) |
| 43 | \( 1 - 448 T + p^{3} T^{2} \) |
| 47 | \( 1 - 149 T + p^{3} T^{2} \) |
| 53 | \( 1 + 672 T + p^{3} T^{2} \) |
| 59 | \( 1 + 615 T + p^{3} T^{2} \) |
| 61 | \( 1 - 322 T + p^{3} T^{2} \) |
| 67 | \( 1 + 411 T + p^{3} T^{2} \) |
| 71 | \( 1 + 968 T + p^{3} T^{2} \) |
| 73 | \( 1 + 227 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 + 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1736 T + p^{3} T^{2} \) |
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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
−10.07818603625037862344403464314, −9.510242112133084318107628041229, −8.546292614276659699751862668490, −7.30660151426419178888294513979, −6.44756272020020946374057673546, −5.71344323135191207491228897788, −4.47076411339765135929645558539, −2.74673496051168261504932207912, −1.54806540767198643707633200614, 0,
1.54806540767198643707633200614, 2.74673496051168261504932207912, 4.47076411339765135929645558539, 5.71344323135191207491228897788, 6.44756272020020946374057673546, 7.30660151426419178888294513979, 8.546292614276659699751862668490, 9.510242112133084318107628041229, 10.07818603625037862344403464314
