| L(s) = 1 | + (−0.0678 + 3.99i)2-s + (−15.9 − 0.542i)4-s + (−16.6 − 28.7i)5-s + (39.9 + 23.0i)7-s + (3.25 − 63.9i)8-s + (116. − 64.4i)10-s + (63.6 + 36.7i)11-s + (151. + 262. i)13-s + (−95.0 + 158. i)14-s + (255. + 17.3i)16-s + 182.·17-s − 314. i·19-s + (250. + 469. i)20-s + (−151. + 251. i)22-s + (290. − 167. i)23-s + ⋯ |
| L(s) = 1 | + (−0.0169 + 0.999i)2-s + (−0.999 − 0.0339i)4-s + (−0.664 − 1.15i)5-s + (0.815 + 0.471i)7-s + (0.0508 − 0.998i)8-s + (1.16 − 0.644i)10-s + (0.525 + 0.303i)11-s + (0.896 + 1.55i)13-s + (−0.484 + 0.807i)14-s + (0.997 + 0.0677i)16-s + 0.629·17-s − 0.870i·19-s + (0.625 + 1.17i)20-s + (−0.312 + 0.520i)22-s + (0.549 − 0.317i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.36324 + 0.793433i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36324 + 0.793433i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0678 - 3.99i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (16.6 + 28.7i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (-39.9 - 23.0i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-63.6 - 36.7i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-151. - 262. i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 - 182.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 314. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-290. + 167. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (357. - 618. i)T + (-3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-985. + 568. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.00e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-557. - 965. i)T + (-1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-2.18e3 - 1.25e3i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (980. + 566. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 - 1.05e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (878. - 507. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (430. - 745. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-559. + 322. i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 9.56e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.89e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (6.76e3 + 3.90e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-7.05e3 - 4.07e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + 7.65e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (6.36e3 - 1.10e4i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
| 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
−13.29884387952047465020500359318, −12.21107385185110993551703938952, −11.31344453100092066092574309663, −9.316685714942198721442394647309, −8.746114246434679158721417549033, −7.76447617346977358412945666101, −6.41596090557727148510567079805, −4.94247330685314476725040561906, −4.18311904255276478535197630988, −1.13409779500929128504192943748,
1.02028424773004737401682230486, 3.04589786265483846174097865661, 4.00257009168146121355745627703, 5.74036479620076693075357074052, 7.57714222541508393558869602371, 8.378350938481854054796116560817, 10.04733039719181573096635941046, 10.88614834248582740830076984457, 11.44921063763272024520946545544, 12.62815963764482562489211041133
