| L(s) = 1 | − 2.44i·2-s + 10·4-s + (−5 + 24.4i)5-s + (35 + 34.2i)7-s − 63.6i·8-s + (59.9 + 12.2i)10-s − 89·11-s − 5·13-s + (84 − 85.7i)14-s + 4.00·16-s + 485·17-s − 220. i·19-s + (−50 + 244. i)20-s + 218. i·22-s + 700. i·23-s + ⋯ |
| L(s) = 1 | − 0.612i·2-s + 0.625·4-s + (−0.200 + 0.979i)5-s + (0.714 + 0.699i)7-s − 0.995i·8-s + (0.599 + 0.122i)10-s − 0.735·11-s − 0.0295·13-s + (0.428 − 0.437i)14-s + 0.0156·16-s + 1.67·17-s − 0.610i·19-s + (−0.125 + 0.612i)20-s + 0.450i·22-s + 1.32i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.391896170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.391896170\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (5 - 24.4i)T \) |
| 7 | \( 1 + (-35 - 34.2i)T \) |
| good | 2 | \( 1 + 2.44iT - 16T^{2} \) |
| 11 | \( 1 + 89T + 1.46e4T^{2} \) |
| 13 | \( 1 + 5T + 2.85e4T^{2} \) |
| 17 | \( 1 - 485T + 8.35e4T^{2} \) |
| 19 | \( 1 + 220. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 700. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 191T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.05e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.63e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.91e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 377. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.19e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.58e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.62e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.93e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.04e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.45e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 8.65e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.56e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.99e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 808. iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.23e3T + 8.85e7T^{2} \) |
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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
−11.22371691712856771712486843299, −10.37086322863805251167560993862, −9.584720415650364060556628506589, −8.031041341440828106490537575904, −7.39537215396424357032380615583, −6.22088072166049808367886720766, −5.14387700811728568326745286728, −3.40687077254302142854468485040, −2.66830794207128654955020219256, −1.39747620591413028060385333409,
0.74709077600556961533737961882, 2.13309939509465209461347071695, 3.88787014104200580428403889628, 5.15049080229915828291103338057, 5.85226312225560475472700200945, 7.40238246850535131142770416461, 7.82564116638020048246516730010, 8.707802842062288920647252913810, 10.13786650879898393081266750385, 10.87960615699059876525022925529
