| L(s) = 1 | + (−1.66 − 2.28i)2-s + (3.38 + 10.4i)3-s + (−2.47 + 7.60i)4-s + (17.3 + 12.5i)5-s + (18.2 − 25.0i)6-s + (27.3 + 8.88i)7-s + (21.5 − 6.99i)8-s + (−31.6 + 23.0i)9-s − 60.5i·10-s + (−102. + 64.5i)11-s − 87.6·12-s + (−109. − 150. i)13-s + (−25.1 − 77.3i)14-s + (−72.4 + 223. i)15-s + (−51.7 − 37.6i)16-s + (158. − 217. i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (0.376 + 1.15i)3-s + (−0.154 + 0.475i)4-s + (0.692 + 0.503i)5-s + (0.506 − 0.696i)6-s + (0.558 + 0.181i)7-s + (0.336 − 0.109i)8-s + (−0.391 + 0.284i)9-s − 0.605i·10-s + (−0.845 + 0.533i)11-s − 0.608·12-s + (−0.645 − 0.888i)13-s + (−0.128 − 0.394i)14-s + (−0.322 + 0.991i)15-s + (−0.202 − 0.146i)16-s + (0.547 − 0.754i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.20348 + 0.393906i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20348 + 0.393906i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.66 + 2.28i)T \) |
| 11 | \( 1 + (102. - 64.5i)T \) |
| good | 3 | \( 1 + (-3.38 - 10.4i)T + (-65.5 + 47.6i)T^{2} \) |
| 5 | \( 1 + (-17.3 - 12.5i)T + (193. + 594. i)T^{2} \) |
| 7 | \( 1 + (-27.3 - 8.88i)T + (1.94e3 + 1.41e3i)T^{2} \) |
| 13 | \( 1 + (109. + 150. i)T + (-8.82e3 + 2.71e4i)T^{2} \) |
| 17 | \( 1 + (-158. + 217. i)T + (-2.58e4 - 7.94e4i)T^{2} \) |
| 19 | \( 1 + (-618. + 201. i)T + (1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 + 644.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (505. + 164. i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 31 | \( 1 + (-1.18e3 + 863. i)T + (2.85e5 - 8.78e5i)T^{2} \) |
| 37 | \( 1 + (367. - 1.12e3i)T + (-1.51e6 - 1.10e6i)T^{2} \) |
| 41 | \( 1 + (2.14e3 - 696. i)T + (2.28e6 - 1.66e6i)T^{2} \) |
| 43 | \( 1 - 1.42e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-409. - 1.25e3i)T + (-3.94e6 + 2.86e6i)T^{2} \) |
| 53 | \( 1 + (-1.89e3 + 1.37e3i)T + (2.43e6 - 7.50e6i)T^{2} \) |
| 59 | \( 1 + (1.01e3 - 3.12e3i)T + (-9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 + (-2.04e3 + 2.81e3i)T + (-4.27e6 - 1.31e7i)T^{2} \) |
| 67 | \( 1 - 1.55e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (3.99e3 + 2.90e3i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (3.56e3 + 1.15e3i)T + (2.29e7 + 1.66e7i)T^{2} \) |
| 79 | \( 1 + (2.63e3 + 3.62e3i)T + (-1.20e7 + 3.70e7i)T^{2} \) |
| 83 | \( 1 + (2.11e3 - 2.91e3i)T + (-1.46e7 - 4.51e7i)T^{2} \) |
| 89 | \( 1 - 9.59e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (1.03e4 - 7.55e3i)T + (2.73e7 - 8.41e7i)T^{2} \) |
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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
−17.66713862411543350490566345662, −16.02075749162925975033652878815, −14.93553959726584570557344545415, −13.63128192128412764418629275141, −11.78807312517808102207765853141, −10.13622759233440482635679442832, −9.754007312412771029030915286116, −7.83684934494090845162262758702, −5.03088502405639105164031106135, −2.83075108727346017896975635341,
1.60500214129071352138658893838, 5.52454523381927311644732304081, 7.31404049235551476448592852484, 8.388748500202573781500003299253, 10.02819391189421950331032921753, 12.09992580929637120189009873234, 13.57237910700832290360693529813, 14.22022919674244182769324637337, 16.09334042786587147317408251277, 17.30754066371927228950325130095
