| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s + 17.1·7-s + 8·8-s + 10·10-s + 47.5·11-s − 12.5·13-s + 34.3·14-s + 16·16-s − 123.·17-s + 65.9·19-s + 20·20-s + 95.1·22-s + 181.·23-s + 25·25-s − 25.0·26-s + 68.6·28-s + 170.·29-s + 19.8·31-s + 32·32-s − 246.·34-s + 85.8·35-s − 27.2·37-s + 131.·38-s + 40·40-s − 172.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.926·7-s + 0.353·8-s + 0.316·10-s + 1.30·11-s − 0.266·13-s + 0.655·14-s + 0.250·16-s − 1.76·17-s + 0.796·19-s + 0.223·20-s + 0.922·22-s + 1.64·23-s + 0.200·25-s − 0.188·26-s + 0.463·28-s + 1.08·29-s + 0.115·31-s + 0.176·32-s − 1.24·34-s + 0.414·35-s − 0.120·37-s + 0.563·38-s + 0.158·40-s − 0.656·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.400744340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.400744340\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 17.1T + 343T^{2} \) |
| 11 | \( 1 - 47.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 170.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 19.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 27.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 172.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 276.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 548.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 670.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 107.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 896.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 807.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 416.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 843.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 983.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 111.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 135.T + 9.12e5T^{2} \) |
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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
−9.854236795401271995640674465469, −9.000283204818785577195294664918, −8.195902054235912042670536165511, −6.85288790106442133106756688156, −6.55080477288104078978003305037, −5.11605839421456370660276338239, −4.67238184083870327413801107069, −3.45486600850490984795400454158, −2.20379222792637822805658460049, −1.16747787302029822436520771987,
1.16747787302029822436520771987, 2.20379222792637822805658460049, 3.45486600850490984795400454158, 4.67238184083870327413801107069, 5.11605839421456370660276338239, 6.55080477288104078978003305037, 6.85288790106442133106756688156, 8.195902054235912042670536165511, 9.000283204818785577195294664918, 9.854236795401271995640674465469
