| L(s) = 1 | + (−139. + 80.5i)2-s + (−619. − 1.07e3i)3-s + (8.86e3 − 1.53e4i)4-s − 4.02e4i·5-s + (1.72e5 + 9.96e4i)6-s + (−4.34e3 − 2.50e3i)7-s + 1.53e6i·8-s + (3.06e4 − 5.31e4i)9-s + (3.23e6 + 5.60e6i)10-s + (3.65e6 − 2.11e6i)11-s − 2.19e7·12-s + (7.65e6 − 1.56e7i)13-s + 8.07e5·14-s + (−4.31e7 + 2.48e7i)15-s + (−5.11e7 − 8.85e7i)16-s + (7.73e7 − 1.33e8i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 + 0.889i)2-s + (−0.490 − 0.849i)3-s + (1.08 − 1.87i)4-s − 1.15i·5-s + (1.51 + 0.872i)6-s + (−0.0139 − 0.00805i)7-s + 2.07i·8-s + (0.0192 − 0.0333i)9-s + (1.02 + 1.77i)10-s + (0.622 − 0.359i)11-s − 2.12·12-s + (0.439 − 0.898i)13-s + 0.0286·14-s + (−0.977 + 0.564i)15-s + (−0.761 − 1.31i)16-s + (0.777 − 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.159032 - 0.534055i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.159032 - 0.534055i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-7.65e6 + 1.56e7i)T \) |
| good | 2 | \( 1 + (139. - 80.5i)T + (4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (619. + 1.07e3i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + 4.02e4iT - 1.22e9T^{2} \) |
| 7 | \( 1 + (4.34e3 + 2.50e3i)T + (4.84e10 + 8.39e10i)T^{2} \) |
| 11 | \( 1 + (-3.65e6 + 2.11e6i)T + (1.72e13 - 2.98e13i)T^{2} \) |
| 17 | \( 1 + (-7.73e7 + 1.33e8i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (2.31e8 + 1.33e8i)T + (2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-4.37e8 - 7.57e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 + (-4.41e8 - 7.65e8i)T + (-5.13e18 + 8.88e18i)T^{2} \) |
| 31 | \( 1 + 8.91e9iT - 2.44e19T^{2} \) |
| 37 | \( 1 + (1.44e10 - 8.32e9i)T + (1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + (1.39e10 - 8.07e9i)T + (4.62e20 - 8.01e20i)T^{2} \) |
| 43 | \( 1 + (1.57e10 - 2.72e10i)T + (-8.59e20 - 1.48e21i)T^{2} \) |
| 47 | \( 1 - 7.32e10iT - 5.46e21T^{2} \) |
| 53 | \( 1 - 3.00e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + (-1.50e11 - 8.66e10i)T + (5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-4.07e10 + 7.06e10i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (8.93e10 - 5.15e10i)T + (2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + (-1.42e11 - 8.22e10i)T + (5.82e23 + 1.00e24i)T^{2} \) |
| 73 | \( 1 - 1.77e12iT - 1.67e24T^{2} \) |
| 79 | \( 1 + 1.38e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.69e12iT - 8.87e24T^{2} \) |
| 89 | \( 1 + (-2.18e12 + 1.25e12i)T + (1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + (9.02e12 + 5.21e12i)T + (3.36e25 + 5.82e25i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.60199504936292815772362348264, −15.29437507060181965211076692836, −13.12948140380869167879564354136, −11.51296820601429420426678322043, −9.568527160667749594021574996549, −8.377989033279130101004865910635, −7.01730596092207695266483974841, −5.61560017144418189061745509665, −1.20595916265728355544599125171, −0.48219834582602822357519030172,
1.82246126783194859754032048600, 3.72026901589203974286716677518, 6.78895155107732751530222619851, 8.681380326317998378298161012684, 10.35136532872419183992795088316, 10.66543373736807396606431731816, 12.12341429264743201414356181160, 14.74439768006014199059829658089, 16.42165179958298684048568842401, 17.28295443176331718998130709073
