| L(s) = 1 | + (−15.5 − 4.16i)2-s + (11.3 + 19.6i)3-s + (2.72 + 1.57i)4-s + (−311. + 311. i)5-s + (−94.7 − 353. i)6-s + (4.36e3 − 1.16e3i)7-s + (2.87e3 + 2.87e3i)8-s + (3.02e3 − 5.23e3i)9-s + (6.13e3 − 3.54e3i)10-s + (−421. + 1.57e3i)11-s + 71.5i·12-s + (1.18e4 + 2.59e4i)13-s − 7.27e4·14-s + (−9.67e3 − 2.59e3i)15-s + (−3.31e4 − 5.74e4i)16-s + (7.52e4 + 4.34e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.971 − 0.260i)2-s + (0.140 + 0.243i)3-s + (0.0106 + 0.00614i)4-s + (−0.498 + 0.498i)5-s + (−0.0731 − 0.272i)6-s + (1.81 − 0.487i)7-s + (0.702 + 0.702i)8-s + (0.460 − 0.797i)9-s + (0.613 − 0.354i)10-s + (−0.0287 + 0.107i)11-s + 0.00345i·12-s + (0.415 + 0.909i)13-s − 1.89·14-s + (−0.191 − 0.0511i)15-s + (−0.506 − 0.876i)16-s + (0.901 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.04649 - 0.0636183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04649 - 0.0636183i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.18e4 - 2.59e4i)T \) |
| good | 2 | \( 1 + (15.5 + 4.16i)T + (221. + 128i)T^{2} \) |
| 3 | \( 1 + (-11.3 - 19.6i)T + (-3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (311. - 311. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + (-4.36e3 + 1.16e3i)T + (4.99e6 - 2.88e6i)T^{2} \) |
| 11 | \( 1 + (421. - 1.57e3i)T + (-1.85e8 - 1.07e8i)T^{2} \) |
| 17 | \( 1 + (-7.52e4 - 4.34e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (3.34e4 + 1.24e5i)T + (-1.47e10 + 8.49e9i)T^{2} \) |
| 23 | \( 1 + (-3.23e4 + 1.86e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-3.41e5 - 5.90e5i)T + (-2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-8.84e5 + 8.84e5i)T - 8.52e11iT^{2} \) |
| 37 | \( 1 + (-4.07e5 + 1.52e6i)T + (-3.04e12 - 1.75e12i)T^{2} \) |
| 41 | \( 1 + (3.34e6 + 8.95e5i)T + (6.91e12 + 3.99e12i)T^{2} \) |
| 43 | \( 1 + (2.56e5 + 1.47e5i)T + (5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-4.49e6 - 4.49e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + 3.91e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (1.32e7 - 3.54e6i)T + (1.27e14 - 7.34e13i)T^{2} \) |
| 61 | \( 1 + (9.16e6 - 1.58e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (7.68e6 + 2.05e6i)T + (3.51e14 + 2.03e14i)T^{2} \) |
| 71 | \( 1 + (-4.41e6 - 1.64e7i)T + (-5.59e14 + 3.22e14i)T^{2} \) |
| 73 | \( 1 + (1.60e7 + 1.60e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + 4.09e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-1.41e6 + 1.41e6i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 + (-2.14e7 + 8.00e7i)T + (-3.40e15 - 1.96e15i)T^{2} \) |
| 97 | \( 1 + (6.58e6 + 2.45e7i)T + (-6.78e15 + 3.91e15i)T^{2} \) |
| 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
−18.05757047919961341694210835954, −17.06451932656818359423853284880, −15.10150267653917132534894606007, −14.07870997040238342822251563329, −11.57516865966254508752155627552, −10.54082159685956665227959415483, −8.869128576425238889872726378972, −7.49628038793065709885923893644, −4.42692411800147895340926471561, −1.29434735113325264811718656737,
1.26379888756999745403560792445, 4.81118622376530704751135954965, 7.948225091333306632398760848738, 8.275971675180415764577441623482, 10.41498642969795495882830176446, 12.12952395399051200782647275103, 13.85398893286762671804544217960, 15.56562987950251745842803676034, 16.89013931372182635484451134630, 18.10240108213011891000786160517
