| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−12.9 + 9.40i)4-s + (28.7 + 9.35i)5-s + (56.7 + 78.0i)7-s + (51.7 + 37.6i)8-s − 121. i·10-s + (−339. − 214. i)11-s + (804. − 261. i)13-s + (226. − 312. i)14-s + (79.1 − 243. i)16-s + (−171. + 529. i)17-s + (−1.64e3 + 2.26e3i)19-s + (−460. + 149. i)20-s + (−397. + 1.55e3i)22-s + 937. i·23-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.514 + 0.167i)5-s + (0.437 + 0.602i)7-s + (0.286 + 0.207i)8-s − 0.382i·10-s + (−0.844 − 0.534i)11-s + (1.32 − 0.428i)13-s + (0.309 − 0.425i)14-s + (0.0772 − 0.237i)16-s + (−0.144 + 0.444i)17-s + (−1.04 + 1.44i)19-s + (−0.257 + 0.0836i)20-s + (−0.175 + 0.685i)22-s + 0.369i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.581903235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.581903235\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (339. + 214. i)T \) |
| good | 5 | \( 1 + (-28.7 - 9.35i)T + (2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-56.7 - 78.0i)T + (-5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-804. + 261. i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (171. - 529. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (1.64e3 - 2.26e3i)T + (-7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 - 937. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-3.23e3 + 2.34e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-2.39e3 - 7.36e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (7.15e3 - 5.20e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-3.91e3 - 2.84e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 2.37e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.34e4 + 1.84e4i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.21e4 - 3.94e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.33e4 - 3.21e4i)T + (-2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.92e4 - 9.51e3i)T + (6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 - 1.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-3.55e4 - 1.15e4i)T + (1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-4.15e4 - 5.71e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (8.56e4 - 2.78e4i)T + (2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.03e4 + 6.27e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 1.16e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.79e3 - 8.58e3i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
| 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
−11.60320009495881401963511030144, −10.61322354046769232042054190217, −10.05068991060316979107152107693, −8.466309201766927287682089884291, −8.259341156520327862351097061543, −6.30010913431034282483404447571, −5.41165640616481396008208450808, −3.84333950952743208314564694246, −2.53083983491818860976054379884, −1.31395500894335649541785970509,
0.55586909625160655742953109947, 2.12960439945188414343918065989, 4.13852728693886457990399247048, 5.17839676360719497555430777241, 6.39361715290828904914013452221, 7.35108460833200144347716703846, 8.463726130084924597303455434462, 9.315391848323356584826295300407, 10.49624943709609554282808577605, 11.19705516150821888800282228201
