| L(s) = 1 | + (6.11 + 10.5i)2-s + (−32.6 − 56.5i)3-s + (−10.7 + 18.6i)4-s − 72.0·5-s + (399. − 691. i)6-s + (171.5 − 297. i)7-s + 1.30e3·8-s + (−1.03e3 + 1.79e3i)9-s + (−440. − 762. i)10-s + (−1.73e3 − 2.99e3i)11-s + 1.40e3·12-s + (6.75e3 − 4.13e3i)13-s + 4.19e3·14-s + (2.35e3 + 4.07e3i)15-s + (9.33e3 + 1.61e4i)16-s + (−2.81e3 + 4.88e3i)17-s + ⋯ |
| L(s) = 1 | + (0.540 + 0.935i)2-s + (−0.697 − 1.20i)3-s + (−0.0840 + 0.145i)4-s − 0.257·5-s + (0.754 − 1.30i)6-s + (0.188 − 0.327i)7-s + 0.899·8-s + (−0.474 + 0.821i)9-s + (−0.139 − 0.241i)10-s + (−0.392 − 0.679i)11-s + 0.234·12-s + (0.853 − 0.521i)13-s + 0.408·14-s + (0.179 + 0.311i)15-s + (0.569 + 0.987i)16-s + (−0.139 + 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.480331 - 1.10934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.480331 - 1.10934i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-171.5 + 297. i)T \) |
| 13 | \( 1 + (-6.75e3 + 4.13e3i)T \) |
| good | 2 | \( 1 + (-6.11 - 10.5i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (32.6 + 56.5i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 72.0T + 7.81e4T^{2} \) |
| 11 | \( 1 + (1.73e3 + 2.99e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (2.81e3 - 4.88e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.69e4 - 2.94e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (4.25e4 + 7.36e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (8.65e4 + 1.49e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (7.94e4 + 1.37e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.85e5 - 3.21e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.48e5 - 2.57e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 8.38e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.12e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.63e5 - 2.83e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.82e5 - 8.35e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.39e6 + 4.15e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (8.52e5 - 1.47e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 3.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.68e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.11e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (2.71e6 + 4.69e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-3.42e6 + 5.93e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
| 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
−12.61059600789976571763161865464, −11.32924081320388272053250199454, −10.45182501249988668054665489262, −8.189280444322645386708537042731, −7.54124332494491177752397174874, −6.19052751522586736044600182961, −5.82691350814353899876104755836, −4.09248833748968806960083260687, −1.72356150116281943829769912280, −0.33929764500987630206928088773,
1.91288208067843572351773183796, 3.60039303666371040930663764953, 4.47561701531428738033813382729, 5.53723190936700933110473916849, 7.37964172036220221440366375010, 9.075116152471277628655329155123, 10.23315489449254644355914149058, 11.13679657674031306757988200386, 11.66459100225413543554707549462, 12.81446371217171269449499953049
