| L(s) = 1 | + (18.3 + 21.8i)3-s + (−6.18 − 2.25i)5-s + (237. − 411. i)7-s + (−14.9 + 84.9i)9-s + (−392. − 679. i)11-s + (2.32e3 − 2.77e3i)13-s + (−64.2 − 176. i)15-s + (1.03e3 + 5.84e3i)17-s + (6.81e3 + 742. i)19-s + (1.33e4 − 2.35e3i)21-s + (−1.87e4 + 6.81e3i)23-s + (−1.19e4 − 1.00e4i)25-s + (1.58e4 − 9.17e3i)27-s + (3.67e4 + 6.48e3i)29-s + (3.74e4 + 2.16e4i)31-s + ⋯ |
| L(s) = 1 | + (0.679 + 0.810i)3-s + (−0.0494 − 0.0180i)5-s + (0.693 − 1.20i)7-s + (−0.0205 + 0.116i)9-s + (−0.294 − 0.510i)11-s + (1.05 − 1.26i)13-s + (−0.0190 − 0.0522i)15-s + (0.209 + 1.18i)17-s + (0.994 + 0.108i)19-s + (1.44 − 0.254i)21-s + (−1.53 + 0.560i)23-s + (−0.763 − 0.641i)25-s + (0.807 − 0.466i)27-s + (1.50 + 0.265i)29-s + (1.25 + 0.726i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.51878 - 0.329082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.51878 - 0.329082i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-6.81e3 - 742. i)T \) |
| good | 3 | \( 1 + (-18.3 - 21.8i)T + (-126. + 717. i)T^{2} \) |
| 5 | \( 1 + (6.18 + 2.25i)T + (1.19e4 + 1.00e4i)T^{2} \) |
| 7 | \( 1 + (-237. + 411. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (392. + 679. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-2.32e3 + 2.77e3i)T + (-8.38e5 - 4.75e6i)T^{2} \) |
| 17 | \( 1 + (-1.03e3 - 5.84e3i)T + (-2.26e7 + 8.25e6i)T^{2} \) |
| 23 | \( 1 + (1.87e4 - 6.81e3i)T + (1.13e8 - 9.51e7i)T^{2} \) |
| 29 | \( 1 + (-3.67e4 - 6.48e3i)T + (5.58e8 + 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-3.74e4 - 2.16e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 4.98e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (4.10e4 + 4.88e4i)T + (-8.24e8 + 4.67e9i)T^{2} \) |
| 43 | \( 1 + (-3.39e4 - 1.23e4i)T + (4.84e9 + 4.06e9i)T^{2} \) |
| 47 | \( 1 + (-1.21e4 + 6.90e4i)T + (-1.01e10 - 3.68e9i)T^{2} \) |
| 53 | \( 1 + (-5.86e4 - 1.61e5i)T + (-1.69e10 + 1.42e10i)T^{2} \) |
| 59 | \( 1 + (3.71e5 - 6.54e4i)T + (3.96e10 - 1.44e10i)T^{2} \) |
| 61 | \( 1 + (8.59e4 - 3.12e4i)T + (3.94e10 - 3.31e10i)T^{2} \) |
| 67 | \( 1 + (-1.04e5 - 1.85e4i)T + (8.50e10 + 3.09e10i)T^{2} \) |
| 71 | \( 1 + (-3.77e4 + 1.03e5i)T + (-9.81e10 - 8.23e10i)T^{2} \) |
| 73 | \( 1 + (4.92e5 - 4.13e5i)T + (2.62e10 - 1.49e11i)T^{2} \) |
| 79 | \( 1 + (4.37e4 + 5.20e4i)T + (-4.22e10 + 2.39e11i)T^{2} \) |
| 83 | \( 1 + (-1.84e5 + 3.20e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-3.38e4 + 4.02e4i)T + (-8.62e10 - 4.89e11i)T^{2} \) |
| 97 | \( 1 + (8.91e5 - 1.57e5i)T + (7.82e11 - 2.84e11i)T^{2} \) |
| 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
−13.72238323457949470433081060763, −12.11937612098379363166080793581, −10.56298397161069941622579225285, −10.20027431377062952468818661339, −8.504289814136458320854035170878, −7.82803030514712166454791366737, −5.92493631823896585943649197259, −4.23802309183706367359145510819, −3.31816920187427068772656624806, −1.02569383682478127152105117948,
1.57068368126621096713219687622, 2.67086097562946225814755691167, 4.74924508485950779592365368794, 6.32481265352833856735497108023, 7.73902660759902018300249091861, 8.543392173213428709871767039898, 9.725718768440631713814765476210, 11.56536113183535863931545936913, 12.10090980783297614027179947816, 13.66494206578857134584538464114
