| L(s) = 1 | + (−19.7 − 60.8i)2-s + (1.97e3 + 1.43e3i)3-s + (−3.31e3 + 2.40e3i)4-s + (−9.03e3 + 2.77e4i)5-s + (4.82e4 − 1.48e5i)6-s + (−1.14e5 + 8.32e4i)7-s + (2.12e5 + 1.54e5i)8-s + (1.34e6 + 4.14e6i)9-s + 1.87e6·10-s + (−5.84e6 + 6.28e5i)11-s − 9.99e6·12-s + (−1.48e6 − 4.57e6i)13-s + (7.33e6 + 5.32e6i)14-s + (−5.76e7 + 4.19e7i)15-s + (5.18e6 − 1.59e7i)16-s + (2.81e7 − 8.65e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (1.56 + 1.13i)3-s + (−0.404 + 0.293i)4-s + (−0.258 + 0.795i)5-s + (0.422 − 1.29i)6-s + (−0.368 + 0.267i)7-s + (0.286 + 0.207i)8-s + (0.845 + 2.60i)9-s + 0.591·10-s + (−0.994 + 0.107i)11-s − 0.966·12-s + (−0.0853 − 0.262i)13-s + (0.260 + 0.189i)14-s + (−1.30 + 0.950i)15-s + (0.0772 − 0.237i)16-s + (0.282 − 0.869i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.813123109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.813123109\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.7 + 60.8i)T \) |
| 11 | \( 1 + (5.84e6 - 6.28e5i)T \) |
| good | 3 | \( 1 + (-1.97e3 - 1.43e3i)T + (4.92e5 + 1.51e6i)T^{2} \) |
| 5 | \( 1 + (9.03e3 - 2.77e4i)T + (-9.87e8 - 7.17e8i)T^{2} \) |
| 7 | \( 1 + (1.14e5 - 8.32e4i)T + (2.99e10 - 9.21e10i)T^{2} \) |
| 13 | \( 1 + (1.48e6 + 4.57e6i)T + (-2.45e14 + 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-2.81e7 + 8.65e7i)T + (-8.01e15 - 5.82e15i)T^{2} \) |
| 19 | \( 1 + (2.31e8 + 1.68e8i)T + (1.29e16 + 3.99e16i)T^{2} \) |
| 23 | \( 1 + 5.37e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (4.03e8 - 2.93e8i)T + (3.17e18 - 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-2.59e9 - 7.99e9i)T + (-1.97e19 + 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-1.16e10 + 8.44e9i)T + (7.52e19 - 2.31e20i)T^{2} \) |
| 41 | \( 1 + (-4.05e10 - 2.94e10i)T + (2.85e20 + 8.79e20i)T^{2} \) |
| 43 | \( 1 + 4.14e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (9.11e8 + 6.62e8i)T + (1.68e21 + 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-3.41e10 - 1.05e11i)T + (-2.10e22 + 1.53e22i)T^{2} \) |
| 59 | \( 1 + (1.49e11 - 1.08e11i)T + (3.24e22 - 9.98e22i)T^{2} \) |
| 61 | \( 1 + (1.80e11 - 5.56e11i)T + (-1.30e23 - 9.51e22i)T^{2} \) |
| 67 | \( 1 - 1.20e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-2.18e10 + 6.70e10i)T + (-9.42e23 - 6.84e23i)T^{2} \) |
| 73 | \( 1 + (-1.12e11 + 8.20e10i)T + (5.16e23 - 1.59e24i)T^{2} \) |
| 79 | \( 1 + (6.33e11 + 1.95e12i)T + (-3.77e24 + 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-4.87e11 + 1.49e12i)T + (-7.17e24 - 5.21e24i)T^{2} \) |
| 89 | \( 1 - 2.13e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (3.50e12 + 1.08e13i)T + (-5.44e25 + 3.95e25i)T^{2} \) |
| 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
−15.24412108165615198598016756953, −14.21110184428499774589280367276, −12.99406148282274361794938521558, −10.88003367858627935674615166426, −10.02268302195787572306680575641, −8.858997758198022785327949902307, −7.61751434229540042881475133030, −4.64306841567809578320513768966, −3.13322463791960560188521030052, −2.50380614583169692469791781400,
0.52143537602158427029131740080, 2.07799180310700279825549395065, 3.93456824937498958264042859697, 6.33550058218155323755465012599, 7.87859104474032286110124477420, 8.379777883788532221476372372242, 9.796813642401495329564238050651, 12.55508347070543991366785148785, 13.22033382145682507627431357015, 14.41413120805907778674288346248
