| L(s) = 1 | − 3.73·2-s − 3·3-s + 5.93·4-s − 18.1·5-s + 11.1·6-s + 21.4·7-s + 7.71·8-s + 9·9-s + 67.7·10-s + 11·11-s − 17.8·12-s + 11.6·13-s − 80.2·14-s + 54.4·15-s − 76.2·16-s + 64.7·17-s − 33.5·18-s − 151.·19-s − 107.·20-s − 64.4·21-s − 41.0·22-s + 94.8·23-s − 23.1·24-s + 204.·25-s − 43.4·26-s − 27·27-s + 127.·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 0.577·3-s + 0.741·4-s − 1.62·5-s + 0.761·6-s + 1.16·7-s + 0.340·8-s + 0.333·9-s + 2.14·10-s + 0.301·11-s − 0.428·12-s + 0.248·13-s − 1.53·14-s + 0.937·15-s − 1.19·16-s + 0.923·17-s − 0.439·18-s − 1.82·19-s − 1.20·20-s − 0.669·21-s − 0.397·22-s + 0.859·23-s − 0.196·24-s + 1.63·25-s − 0.328·26-s − 0.192·27-s + 0.860·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 3.73T + 8T^{2} \) |
| 5 | \( 1 + 18.1T + 125T^{2} \) |
| 7 | \( 1 - 21.4T + 343T^{2} \) |
| 13 | \( 1 - 11.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 64.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 151.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 94.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 112.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 242.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 115.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 401.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 649.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 54.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 249.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 27.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.31e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 497.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.53e3T + 9.12e5T^{2} \) |
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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
−8.542675473647519333932614148868, −7.77814677915094165321574964940, −7.21169427138474957967394894761, −6.36816100936902480555677400840, −4.90519091453338329219701629859, −4.47990459969415523465363280197, −3.45108009201053187088821181919, −1.82019737895669907390404855362, −0.908513392336422109562763432184, 0,
0.908513392336422109562763432184, 1.82019737895669907390404855362, 3.45108009201053187088821181919, 4.47990459969415523465363280197, 4.90519091453338329219701629859, 6.36816100936902480555677400840, 7.21169427138474957967394894761, 7.77814677915094165321574964940, 8.542675473647519333932614148868
