L(s) = 1 | + 4.31·2-s − 5·3-s + 10.6·4-s − 5·5-s − 21.5·6-s + 11.3·8-s − 2·9-s − 21.5·10-s + 19.7·11-s − 53.1·12-s − 71.3·13-s + 25·15-s − 36.0·16-s + 31.3·17-s − 8.63·18-s − 136.·19-s − 53.1·20-s + 85.1·22-s − 100.·23-s − 56.8·24-s + 25·25-s − 307.·26-s + 145·27-s − 288.·29-s + 107.·30-s + 208.·31-s − 246.·32-s + ⋯ |
L(s) = 1 | + 1.52·2-s − 0.962·3-s + 1.32·4-s − 0.447·5-s − 1.46·6-s + 0.502·8-s − 0.0740·9-s − 0.682·10-s + 0.540·11-s − 1.27·12-s − 1.52·13-s + 0.430·15-s − 0.562·16-s + 0.447·17-s − 0.113·18-s − 1.64·19-s − 0.594·20-s + 0.825·22-s − 0.914·23-s − 0.483·24-s + 0.200·25-s − 2.32·26-s + 1.03·27-s − 1.84·29-s + 0.656·30-s + 1.21·31-s − 1.36·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 4.31T + 8T^{2} \) |
| 3 | \( 1 + 5T + 27T^{2} \) |
| 11 | \( 1 - 19.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 288.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 181.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 18.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 127.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 322.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 341.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 84.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 315.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 643.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−11.55639106780278446091833485974, −10.73759722190597981294620729364, −9.400656195664871413315169362982, −7.84200990705642548892143387115, −6.60332938451781971481346140290, −5.86149798837209534943728765147, −4.81715488063016183995797723016, −4.00443109216199791205280469390, −2.47739258038033237411274477461, 0,
2.47739258038033237411274477461, 4.00443109216199791205280469390, 4.81715488063016183995797723016, 5.86149798837209534943728765147, 6.60332938451781971481346140290, 7.84200990705642548892143387115, 9.400656195664871413315169362982, 10.73759722190597981294620729364, 11.55639106780278446091833485974