| L(s) = 1 | + (−10.2 + 5.89i)2-s + (−41.7 − 72.2i)3-s + (5.57 − 9.65i)4-s − 264. i·5-s + (852. + 491. i)6-s + (297. + 171.5i)7-s − 1.37e3i·8-s + (−2.38e3 + 4.13e3i)9-s + (1.56e3 + 2.70e3i)10-s + (1.67e3 − 967. i)11-s − 930.·12-s + (−7.87e3 + 823. i)13-s − 4.04e3·14-s + (−1.91e4 + 1.10e4i)15-s + (8.84e3 + 1.53e4i)16-s + (−1.84e4 + 3.19e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.902 + 0.521i)2-s + (−0.891 − 1.54i)3-s + (0.0435 − 0.0754i)4-s − 0.947i·5-s + (1.61 + 0.929i)6-s + (0.327 + 0.188i)7-s − 0.951i·8-s + (−1.09 + 1.88i)9-s + (0.493 + 0.855i)10-s + (0.379 − 0.219i)11-s − 0.155·12-s + (−0.994 + 0.103i)13-s − 0.394·14-s + (−1.46 + 0.844i)15-s + (0.539 + 0.934i)16-s + (−0.911 + 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0911i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.437880 + 0.0199997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.437880 + 0.0199997i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-297. - 171.5i)T \) |
| 13 | \( 1 + (7.87e3 - 823. i)T \) |
| good | 2 | \( 1 + (10.2 - 5.89i)T + (64 - 110. i)T^{2} \) |
| 3 | \( 1 + (41.7 + 72.2i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 264. iT - 7.81e4T^{2} \) |
| 11 | \( 1 + (-1.67e3 + 967. i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.84e4 - 3.19e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.33e4 + 1.34e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.58e4 + 2.73e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-1.10e5 - 1.91e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.13e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (-3.62e4 + 2.09e4i)T + (4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (3.60e5 - 2.08e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.43e5 - 2.49e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + 1.40e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.29e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.66e6 - 9.60e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-7.29e5 + 1.26e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.47e6 + 8.53e5i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (4.41e6 + 2.54e6i)T + (4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 4.81e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 1.42e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.27e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (8.38e6 - 4.84e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-6.77e6 - 3.91e6i)T + (4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.69158736864583278793435372382, −11.89995659953490214004401958084, −10.54961827547282783274909890792, −8.799571222419173276133907419745, −8.242146776466893158114873446448, −7.03294014801559599153510709323, −6.20503728544827501153283001646, −4.68805486601039754704991944026, −1.84802243425547570770476136086, −0.69278380153760156796515710695,
0.35616728802023415257197544588, 2.57595787528219140139975563365, 4.32051037186872382331249710900, 5.37651248478502289443926056710, 6.91930085178397980835468144988, 8.735878500232842512351974119851, 9.926371327832297987189211111991, 10.21827489017484157527950164069, 11.32026232532139950716794079579, 11.78525605479678315645485892716
