| L(s) = 1 | + (5.40 − 20.1i)2-s + (−57.1 − 99.0i)3-s + (−156. − 90.2i)4-s + (54.1 + 54.1i)5-s + (−2.30e3 + 618. i)6-s + (319. + 1.19e3i)7-s + (1.11e3 − 1.11e3i)8-s + (−3.26e3 + 5.64e3i)9-s + (1.38e3 − 800. i)10-s + (−2.09e4 − 5.62e3i)11-s + 2.06e4i·12-s + (8.89e3 − 2.71e4i)13-s + 2.58e4·14-s + (2.26e3 − 8.46e3i)15-s + (−3.95e4 − 6.85e4i)16-s + (3.64e4 + 2.10e4i)17-s + ⋯ |
| L(s) = 1 | + (0.337 − 1.26i)2-s + (−0.706 − 1.22i)3-s + (−0.610 − 0.352i)4-s + (0.0866 + 0.0866i)5-s + (−1.78 + 0.477i)6-s + (0.133 + 0.497i)7-s + (0.272 − 0.272i)8-s + (−0.496 + 0.860i)9-s + (0.138 − 0.0800i)10-s + (−1.43 − 0.384i)11-s + 0.995i·12-s + (0.311 − 0.950i)13-s + 0.672·14-s + (0.0447 − 0.167i)15-s + (−0.604 − 1.04i)16-s + (0.436 + 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.187124 + 1.33649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.187124 + 1.33649i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-8.89e3 + 2.71e4i)T \) |
| good | 2 | \( 1 + (-5.40 + 20.1i)T + (-221. - 128i)T^{2} \) |
| 3 | \( 1 + (57.1 + 99.0i)T + (-3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-54.1 - 54.1i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 + (-319. - 1.19e3i)T + (-4.99e6 + 2.88e6i)T^{2} \) |
| 11 | \( 1 + (2.09e4 + 5.62e3i)T + (1.85e8 + 1.07e8i)T^{2} \) |
| 17 | \( 1 + (-3.64e4 - 2.10e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.68e5 + 4.51e4i)T + (1.47e10 - 8.49e9i)T^{2} \) |
| 23 | \( 1 + (1.49e5 - 8.61e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-1.20e5 - 2.09e5i)T + (-2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-3.20e5 - 3.20e5i)T + 8.52e11iT^{2} \) |
| 37 | \( 1 + (-3.21e6 - 8.61e5i)T + (3.04e12 + 1.75e12i)T^{2} \) |
| 41 | \( 1 + (-1.23e6 + 4.60e6i)T + (-6.91e12 - 3.99e12i)T^{2} \) |
| 43 | \( 1 + (-1.14e6 - 6.61e5i)T + (5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (4.28e6 - 4.28e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 - 1.21e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + (9.14e5 + 3.41e6i)T + (-1.27e14 + 7.34e13i)T^{2} \) |
| 61 | \( 1 + (1.01e7 - 1.76e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-3.69e6 + 1.38e7i)T + (-3.51e14 - 2.03e14i)T^{2} \) |
| 71 | \( 1 + (1.26e6 - 3.40e5i)T + (5.59e14 - 3.22e14i)T^{2} \) |
| 73 | \( 1 + (3.03e7 - 3.03e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 1.44e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (2.29e7 + 2.29e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 + (2.50e6 + 6.70e5i)T + (3.40e15 + 1.96e15i)T^{2} \) |
| 97 | \( 1 + (4.15e7 - 1.11e7i)T + (6.78e15 - 3.91e15i)T^{2} \) |
| 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
−17.99517003100191501365061325876, −15.95180490742217166759287492168, −13.58856931125828552413767239353, −12.66482584806032510074378260231, −11.67119916271966793285849763772, −10.38412824675311736457878363199, −7.70550174445848981230850283231, −5.61307349762683271397408070115, −2.67187666897365685995212959993, −0.841458220305917052577332841642,
4.50796877600453094383657004175, 5.68588300071604713550302223016, 7.62437390874356697307467222893, 9.868479059730432454460339514995, 11.25721148962171898873871451357, 13.58887013790267977387090227194, 15.01785897899663531824213571613, 16.13998003380288461644003275129, 16.65647741325338451008751252488, 18.11112653609110575003642037029
