| L(s) = 1 | − 1.80·2-s − 3·3-s − 4.75·4-s + 13.7·5-s + 5.40·6-s + 7.13·7-s + 22.9·8-s + 9·9-s − 24.7·10-s − 11·11-s + 14.2·12-s − 57.5·13-s − 12.8·14-s − 41.1·15-s − 3.40·16-s + 37.5·17-s − 16.2·18-s + 59.5·19-s − 65.1·20-s − 21.3·21-s + 19.8·22-s + 121.·23-s − 68.9·24-s + 63.1·25-s + 103.·26-s − 27·27-s − 33.8·28-s + ⋯ |
| L(s) = 1 | − 0.637·2-s − 0.577·3-s − 0.594·4-s + 1.22·5-s + 0.367·6-s + 0.385·7-s + 1.01·8-s + 0.333·9-s − 0.781·10-s − 0.301·11-s + 0.342·12-s − 1.22·13-s − 0.245·14-s − 0.708·15-s − 0.0531·16-s + 0.535·17-s − 0.212·18-s + 0.719·19-s − 0.728·20-s − 0.222·21-s + 0.192·22-s + 1.09·23-s − 0.586·24-s + 0.504·25-s + 0.782·26-s − 0.192·27-s − 0.228·28-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 | \( 1 + 1.80T + 8T^{2} \) |
| 5 | \( 1 - 13.7T + 125T^{2} \) |
| 7 | \( 1 - 7.13T + 343T^{2} \) |
| 13 | \( 1 + 57.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 129.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 126.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 153.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 141.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 128.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 213.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 379.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 35.7T + 2.05e5T^{2} \) |
| 67 | \( 1 + 275.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 146.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 663.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 127.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 803.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 37.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.40e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.506218873765470353229723047191, −7.64271199043892613912976981432, −6.98027253316679782229142217218, −5.89017622492163751517578250157, −5.05729295043210683822936360060, −4.81049091775451903189028764256, −3.27688279213590323580716043382, −1.99786410653681859137674236333, −1.16592920861015003364541629140, 0,
1.16592920861015003364541629140, 1.99786410653681859137674236333, 3.27688279213590323580716043382, 4.81049091775451903189028764256, 5.05729295043210683822936360060, 5.89017622492163751517578250157, 6.98027253316679782229142217218, 7.64271199043892613912976981432, 8.506218873765470353229723047191
