| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 5.58·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 11.1·10-s + 15.6·11-s − 12·12-s − 33.3·13-s − 14·14-s + 16.7·15-s + 16·16-s + 1.50·17-s − 18·18-s − 19·19-s − 22.3·20-s − 21·21-s − 31.2·22-s − 24.5·23-s + 24·24-s − 93.8·25-s + 66.7·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.499·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.353·10-s + 0.428·11-s − 0.288·12-s − 0.712·13-s − 0.267·14-s + 0.288·15-s + 0.250·16-s + 0.0215·17-s − 0.235·18-s − 0.229·19-s − 0.249·20-s − 0.218·21-s − 0.302·22-s − 0.222·23-s + 0.204·24-s − 0.750·25-s + 0.503·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 + 5.58T + 125T^{2} \) |
| 11 | \( 1 - 15.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.50T + 4.91e3T^{2} \) |
| 23 | \( 1 + 24.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 309.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 15.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 27.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 249.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 72.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 443.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 189.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 387.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 281.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 501.T + 9.12e5T^{2} \) |
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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
−9.581163183442073792183082264987, −8.503872491643035417081584155038, −7.81907469260055591080193268494, −6.93316149541306935141688754948, −6.11677446548793850903579901128, −4.95901289309453971382883336792, −4.01702299311503746775969058505, −2.57013875609385840926048253633, −1.22240295459483704594692061918, 0,
1.22240295459483704594692061918, 2.57013875609385840926048253633, 4.01702299311503746775969058505, 4.95901289309453971382883336792, 6.11677446548793850903579901128, 6.93316149541306935141688754948, 7.81907469260055591080193268494, 8.503872491643035417081584155038, 9.581163183442073792183082264987
