L(s) = 1 | + (−4.89 − 2.83i)2-s + (−6.36 + 6.36i)3-s + (15.9 + 27.7i)4-s + (−71.9 − 71.9i)5-s + (49.1 − 13.1i)6-s + 158. i·7-s + (0.326 − 181. i)8-s − 81i·9-s + (148. + 556. i)10-s + (251. + 251. i)11-s + (−278. − 74.8i)12-s + (365. − 365. i)13-s + (449. − 777. i)14-s + 916.·15-s + (−514. + 885. i)16-s + 1.50e3·17-s + ⋯ |
L(s) = 1 | + (−0.865 − 0.500i)2-s + (−0.408 + 0.408i)3-s + (0.498 + 0.866i)4-s + (−1.28 − 1.28i)5-s + (0.557 − 0.149i)6-s + 1.22i·7-s + (0.00180 − 0.999i)8-s − 0.333i·9-s + (0.470 + 1.75i)10-s + (0.627 + 0.627i)11-s + (−0.557 − 0.150i)12-s + (0.600 − 0.600i)13-s + (0.613 − 1.06i)14-s + 1.05·15-s + (−0.502 + 0.864i)16-s + 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.759594 - 0.0507035i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.759594 - 0.0507035i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (4.89 + 2.83i)T \) |
| 3 | \( 1 + (6.36 - 6.36i)T \) |
good | 5 | \( 1 + (71.9 + 71.9i)T + 3.12e3iT^{2} \) |
| 7 | \( 1 - 158. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-251. - 251. i)T + 1.61e5iT^{2} \) |
| 13 | \( 1 + (-365. + 365. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 - 1.50e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-1.79e3 + 1.79e3i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 - 2.09e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (2.34e3 - 2.34e3i)T - 2.05e7iT^{2} \) |
| 31 | \( 1 + 976.T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-4.69e3 - 4.69e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.43e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-3.69e3 - 3.69e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + 4.92e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-5.19e3 - 5.19e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (-1.64e4 - 1.64e4i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (-1.75e4 + 1.75e4i)T - 8.44e8iT^{2} \) |
| 67 | \( 1 + (-2.87e4 + 2.87e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 823. iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 6.77e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 4.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-3.13e4 + 3.13e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 2.43e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 9.82e4T + 8.58e9T^{2} \) |
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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
−15.32111143184462892697127619888, −12.77908078391333826855219491628, −11.99360307432456284275736073698, −11.38014705035528517646707932210, −9.544592143299401336034105744405, −8.751607931419586845753713399903, −7.55348522500824098223200642297, −5.27589603536956556479516978713, −3.57477042559442777119398140588, −0.955246705900861514917751876230,
0.833705461673835665508579021852, 3.68094592674337121663808572000, 6.24254046034457480446385689368, 7.28922165686740386503476460773, 8.018824779375850171931925336261, 10.07092779353512413870585583336, 11.08394147844308287020704985115, 11.78167591100800744144938438447, 14.04170489350671399858004336417, 14.65045078705014753088701477686