L(s) = 1 | + 5.05·2-s − 9.36·3-s + 17.5·4-s − 14.1·5-s − 47.3·6-s + 13.7·7-s + 48.1·8-s + 60.6·9-s − 71.4·10-s + 10.3·11-s − 164.·12-s − 61.4·13-s + 69.2·14-s + 132.·15-s + 103.·16-s − 24.4·17-s + 306.·18-s + 65.5·19-s − 247.·20-s − 128.·21-s + 52.3·22-s + 23.4·23-s − 450.·24-s + 75.0·25-s − 310.·26-s − 315.·27-s + 240.·28-s + ⋯ |
L(s) = 1 | + 1.78·2-s − 1.80·3-s + 2.19·4-s − 1.26·5-s − 3.21·6-s + 0.740·7-s + 2.12·8-s + 2.24·9-s − 2.25·10-s + 0.284·11-s − 3.94·12-s − 1.31·13-s + 1.32·14-s + 2.27·15-s + 1.60·16-s − 0.348·17-s + 4.01·18-s + 0.791·19-s − 2.77·20-s − 1.33·21-s + 0.507·22-s + 0.212·23-s − 3.83·24-s + 0.600·25-s − 2.34·26-s − 2.24·27-s + 1.62·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
good | 2 | \( 1 - 5.05T + 8T^{2} \) |
| 3 | \( 1 + 9.36T + 27T^{2} \) |
| 5 | \( 1 + 14.1T + 125T^{2} \) |
| 7 | \( 1 - 13.7T + 343T^{2} \) |
| 11 | \( 1 - 10.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 23.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 316.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 13.2T + 6.89e4T^{2} \) |
| 47 | \( 1 - 450.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 205.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 334.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 811.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 113.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 53.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 108.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 723.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 191.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 419.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 953.T + 9.12e5T^{2} \) |
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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
−7.917609735440536127368960774886, −7.25170490795463396395263237949, −6.71972105326329944332296622614, −5.75281619224567399787256941819, −5.09457046870330512969421157730, −4.50649233435241707534009024555, −4.03719186770144373843517010925, −2.73339450904125160075828523367, −1.28915991469340185902227339885, 0,
1.28915991469340185902227339885, 2.73339450904125160075828523367, 4.03719186770144373843517010925, 4.50649233435241707534009024555, 5.09457046870330512969421157730, 5.75281619224567399787256941819, 6.71972105326329944332296622614, 7.25170490795463396395263237949, 7.917609735440536127368960774886