| L(s) = 1 | + (−29.9 + 5.27i)2-s + (216. + 109. i)3-s + (−94.2 + 34.3i)4-s + (378. − 450. i)5-s + (−7.06e3 − 2.14e3i)6-s + (1.10e4 + 4.03e3i)7-s + (2.95e4 − 1.70e4i)8-s + (3.48e4 + 4.76e4i)9-s + (−8.94e3 + 1.54e4i)10-s + (−1.14e5 − 1.36e5i)11-s + (−2.41e4 − 2.93e3i)12-s + (6.40e4 − 3.62e5i)13-s + (−3.52e5 − 6.21e4i)14-s + (1.31e5 − 5.60e4i)15-s + (−7.16e5 + 6.01e5i)16-s + (1.77e6 + 1.02e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.935 + 0.164i)2-s + (0.891 + 0.452i)3-s + (−0.0920 + 0.0335i)4-s + (0.121 − 0.144i)5-s + (−0.908 − 0.276i)6-s + (0.658 + 0.239i)7-s + (0.903 − 0.521i)8-s + (0.590 + 0.807i)9-s + (−0.0894 + 0.154i)10-s + (−0.711 − 0.847i)11-s + (−0.0972 − 0.0117i)12-s + (0.172 − 0.977i)13-s + (−0.655 − 0.115i)14-s + (0.173 − 0.0738i)15-s + (−0.683 + 0.573i)16-s + (1.25 + 0.722i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.41602 + 0.700147i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41602 + 0.700147i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-216. - 109. i)T \) |
| good | 2 | \( 1 + (29.9 - 5.27i)T + (962. - 350. i)T^{2} \) |
| 5 | \( 1 + (-378. + 450. i)T + (-1.69e6 - 9.61e6i)T^{2} \) |
| 7 | \( 1 + (-1.10e4 - 4.03e3i)T + (2.16e8 + 1.81e8i)T^{2} \) |
| 11 | \( 1 + (1.14e5 + 1.36e5i)T + (-4.50e9 + 2.55e10i)T^{2} \) |
| 13 | \( 1 + (-6.40e4 + 3.62e5i)T + (-1.29e11 - 4.71e10i)T^{2} \) |
| 17 | \( 1 + (-1.77e6 - 1.02e6i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-3.47e5 - 6.01e5i)T + (-3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-1.96e6 - 5.40e6i)T + (-3.17e13 + 2.66e13i)T^{2} \) |
| 29 | \( 1 + (-3.42e7 + 6.03e6i)T + (3.95e14 - 1.43e14i)T^{2} \) |
| 31 | \( 1 + (9.75e6 - 3.55e6i)T + (6.27e14 - 5.26e14i)T^{2} \) |
| 37 | \( 1 + (3.36e7 - 5.83e7i)T + (-2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 + (-1.59e8 - 2.81e7i)T + (1.26e16 + 4.59e15i)T^{2} \) |
| 43 | \( 1 + (4.92e7 - 4.13e7i)T + (3.75e15 - 2.12e16i)T^{2} \) |
| 47 | \( 1 + (1.76e7 - 4.83e7i)T + (-4.02e16 - 3.38e16i)T^{2} \) |
| 53 | \( 1 + 2.14e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-1.70e8 + 2.03e8i)T + (-8.87e16 - 5.03e17i)T^{2} \) |
| 61 | \( 1 + (3.87e8 + 1.41e8i)T + (5.46e17 + 4.58e17i)T^{2} \) |
| 67 | \( 1 + (-2.93e8 + 1.66e9i)T + (-1.71e18 - 6.23e17i)T^{2} \) |
| 71 | \( 1 + (-8.21e8 - 4.74e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-1.58e9 - 2.74e9i)T + (-2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (2.34e8 + 1.32e9i)T + (-8.89e18 + 3.23e18i)T^{2} \) |
| 83 | \( 1 + (-3.90e9 + 6.89e8i)T + (1.45e19 - 5.30e18i)T^{2} \) |
| 89 | \( 1 + (3.19e8 - 1.84e8i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + (8.51e9 - 7.14e9i)T + (1.28e19 - 7.26e19i)T^{2} \) |
| 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
−15.36026992299440335142985909495, −14.03170206119688689796782610886, −12.93152420218492186980380696511, −10.76864473818985725232144579168, −9.739784001427440683119801974658, −8.368722487621897483046002449455, −7.86240727580553742333927231671, −5.19338459988216614732997640839, −3.31928768634663777925893482256, −1.22535288327387712135661321964,
0.961100786114492413037017451203, 2.33838926318938900990700228495, 4.61422170010074951891945795577, 7.17696020062998008056116483747, 8.219884412212038709151561442209, 9.406726710669779835336130780813, 10.52188919452730767693505070299, 12.28562374169940952355089142285, 13.87383092644030860478328860020, 14.49645766973506408732598112625
