| L(s) = 1 | + 0.628·2-s − 4.58·3-s − 31.6·4-s + 25·5-s − 2.88·6-s − 116.·7-s − 39.9·8-s − 221.·9-s + 15.7·10-s + 145.·12-s + 349.·13-s − 73.0·14-s − 114.·15-s + 986.·16-s − 264.·17-s − 139.·18-s + 1.72e3·19-s − 790.·20-s + 533.·21-s + 2.69e3·23-s + 183.·24-s + 625·25-s + 219.·26-s + 2.13e3·27-s + 3.67e3·28-s − 907.·29-s − 72.0·30-s + ⋯ |
| L(s) = 1 | + 0.111·2-s − 0.294·3-s − 0.987·4-s + 0.447·5-s − 0.0326·6-s − 0.897·7-s − 0.220·8-s − 0.913·9-s + 0.0496·10-s + 0.290·12-s + 0.572·13-s − 0.0996·14-s − 0.131·15-s + 0.963·16-s − 0.222·17-s − 0.101·18-s + 1.09·19-s − 0.441·20-s + 0.264·21-s + 1.06·23-s + 0.0649·24-s + 0.200·25-s + 0.0636·26-s + 0.563·27-s + 0.885·28-s − 0.200·29-s − 0.0146·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.628T + 32T^{2} \) |
| 3 | \( 1 + 4.58T + 243T^{2} \) |
| 7 | \( 1 + 116.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 349.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 264.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.72e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 907.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 515.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 315.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.49e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.57e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.45e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.77e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.23e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.19e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.13e5T + 8.58e9T^{2} \) |
| 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
−9.194546882749711777507395680030, −8.964115855847663693562077930170, −7.72170189467236442651495044372, −6.49750577247889192812816537712, −5.69713838866165377331686344050, −4.97981241720918054460864565024, −3.65755700848920526478450339999, −2.84664189235314222809516594266, −1.06367430225142576018731922263, 0,
1.06367430225142576018731922263, 2.84664189235314222809516594266, 3.65755700848920526478450339999, 4.97981241720918054460864565024, 5.69713838866165377331686344050, 6.49750577247889192812816537712, 7.72170189467236442651495044372, 8.964115855847663693562077930170, 9.194546882749711777507395680030
