| L(s) = 1 | + (31.9 − 31.9i)3-s + (95.0 − 95.0i)5-s + 293.·7-s − 1.31e3i·9-s + (1.13e3 + 1.13e3i)11-s + (300. + 300. i)13-s − 6.07e3i·15-s − 7.15e3·17-s + (−4.87e3 + 4.87e3i)19-s + (9.38e3 − 9.38e3i)21-s + 1.05e4·23-s − 2.43e3i·25-s + (−1.86e4 − 1.86e4i)27-s + (−1.74e4 − 1.74e4i)29-s − 3.95e4i·31-s + ⋯ |
| L(s) = 1 | + (1.18 − 1.18i)3-s + (0.760 − 0.760i)5-s + 0.855·7-s − 1.80i·9-s + (0.851 + 0.851i)11-s + (0.136 + 0.136i)13-s − 1.79i·15-s − 1.45·17-s + (−0.710 + 0.710i)19-s + (1.01 − 1.01i)21-s + 0.866·23-s − 0.155i·25-s + (−0.948 − 0.948i)27-s + (−0.716 − 0.716i)29-s − 1.32i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.46465 - 2.13421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46465 - 2.13421i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-31.9 + 31.9i)T - 729iT^{2} \) |
| 5 | \( 1 + (-95.0 + 95.0i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 - 293.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-1.13e3 - 1.13e3i)T + 1.77e6iT^{2} \) |
| 13 | \( 1 + (-300. - 300. i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 + 7.15e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (4.87e3 - 4.87e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 - 1.05e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.74e4 + 1.74e4i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 + 3.95e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-1.44e3 + 1.44e3i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 - 6.11e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (4.00e3 + 4.00e3i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 - 1.45e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (6.90e4 - 6.90e4i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 + (-6.43e4 - 6.43e4i)T + 4.21e10iT^{2} \) |
| 61 | \( 1 + (-6.16e4 - 6.16e4i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 + (1.92e5 - 1.92e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 - 2.31e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 5.06e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 7.47e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-4.24e4 + 4.24e4i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + 3.87e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.13e6T + 8.32e11T^{2} \) |
| 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
−13.31635383481712434485752443521, −12.78298202937757143079194271266, −11.44331205200164479215347795946, −9.457187154088285466537706811053, −8.713991309675995369331538791765, −7.59770976212554946615414104318, −6.31686433617417704435797828321, −4.38672195321634777535356666213, −2.16924482679639466586540785974, −1.39512714538412261189470572213,
2.14826195681800606329843086010, 3.48638245982897492824873951073, 4.88950390014086239487228627116, 6.71444270035567973758608527537, 8.559625704799695212910426838183, 9.126178782737854085269880895037, 10.53889761621776580433502100712, 11.15697685186624749820109289178, 13.36466512738776871189520537933, 14.23141486032620868603705656756
