| L(s) = 1 | + 3.28·2-s − 3·3-s + 2.79·4-s − 21.6·5-s − 9.85·6-s + 15.7·7-s − 17.0·8-s + 9·9-s − 71.2·10-s − 11·11-s − 8.38·12-s + 11.1·13-s + 51.6·14-s + 65.0·15-s − 78.5·16-s + 39.3·17-s + 29.5·18-s − 4.48·19-s − 60.6·20-s − 47.1·21-s − 36.1·22-s + 178.·23-s + 51.2·24-s + 344.·25-s + 36.6·26-s − 27·27-s + 43.9·28-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 0.577·3-s + 0.349·4-s − 1.93·5-s − 0.670·6-s + 0.848·7-s − 0.755·8-s + 0.333·9-s − 2.25·10-s − 0.301·11-s − 0.201·12-s + 0.238·13-s + 0.985·14-s + 1.11·15-s − 1.22·16-s + 0.561·17-s + 0.387·18-s − 0.0541·19-s − 0.677·20-s − 0.489·21-s − 0.350·22-s + 1.61·23-s + 0.436·24-s + 2.75·25-s + 0.276·26-s − 0.192·27-s + 0.296·28-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 | \( 1 - 3.28T + 8T^{2} \) |
| 5 | \( 1 + 21.6T + 125T^{2} \) |
| 7 | \( 1 - 15.7T + 343T^{2} \) |
| 13 | \( 1 - 11.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4.48T + 6.85e3T^{2} \) |
| 23 | \( 1 - 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 54.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 36.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 49.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 399.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 551.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 260.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 135.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 48.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
| 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
−8.391332320275080189830960407089, −7.38690846669453792765592931815, −6.90360042921992075231103347244, −5.68575573610964981312119927065, −4.91163537645489052439071219572, −4.46560821877316937896282162717, −3.63353089655235501970443373536, −2.89741819572053054035752149315, −1.07789376070128994677252493540, 0,
1.07789376070128994677252493540, 2.89741819572053054035752149315, 3.63353089655235501970443373536, 4.46560821877316937896282162717, 4.91163537645489052439071219572, 5.68575573610964981312119927065, 6.90360042921992075231103347244, 7.38690846669453792765592931815, 8.391332320275080189830960407089
