| L(s) = 1 | + (6.07 − 4.41i)2-s + (2.68 − 8.24i)3-s + (−22.1 + 68.0i)4-s + (2.44 + 279. i)5-s + (−20.1 − 61.9i)6-s − 1.15e3·7-s + (463. + 1.42e3i)8-s + (1.70e3 + 1.24e3i)9-s + (1.24e3 + 1.68e3i)10-s + (348. − 253. i)11-s + (502. + 364. i)12-s + (226. + 164. i)13-s + (−7.01e3 + 5.09e3i)14-s + (2.31e3 + 728. i)15-s + (1.69e3 + 1.23e3i)16-s + (1.01e4 + 3.11e4i)17-s + ⋯ |
| L(s) = 1 | + (0.537 − 0.390i)2-s + (0.0573 − 0.176i)3-s + (−0.172 + 0.531i)4-s + (0.00875 + 0.999i)5-s + (−0.0380 − 0.117i)6-s − 1.27·7-s + (0.319 + 0.984i)8-s + (0.781 + 0.567i)9-s + (0.394 + 0.533i)10-s + (0.0789 − 0.0573i)11-s + (0.0838 + 0.0609i)12-s + (0.0286 + 0.0208i)13-s + (−0.682 + 0.496i)14-s + (0.176 + 0.0557i)15-s + (0.103 + 0.0753i)16-s + (0.499 + 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.31377 + 1.10635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31377 + 1.10635i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.44 - 279. i)T \) |
| good | 2 | \( 1 + (-6.07 + 4.41i)T + (39.5 - 121. i)T^{2} \) |
| 3 | \( 1 + (-2.68 + 8.24i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 7 | \( 1 + 1.15e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + (-348. + 253. i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-226. - 164. i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-1.01e4 - 3.11e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (9.48e3 + 2.92e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-3.59e3 + 2.60e3i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-3.43e4 + 1.05e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (4.48e4 + 1.37e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-3.91e5 - 2.84e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-4.19e5 - 3.05e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 - 1.46e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (3.61e5 - 1.11e6i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-2.76e5 + 8.50e5i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-2.09e6 - 1.52e6i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-2.34e6 + 1.70e6i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-1.95e5 - 6.00e5i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-1.22e6 + 3.77e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (1.30e5 - 9.46e4i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (1.13e6 - 3.50e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (9.85e5 + 3.03e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (-5.36e6 + 3.89e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (2.38e6 - 7.32e6i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−16.23756137840356162446753920300, −14.84886761419758692042091559156, −13.37495768931236370788348395542, −12.77171147306200407230266506639, −11.18604128753733167700053756354, −9.870607488328610283293347634763, −7.85861171600230086695416363881, −6.38679565995578285379270572842, −4.00495200538216726539207765530, −2.60000525823433525221568353080,
0.78235907385636729868394376730, 3.92446069409103553508783546472, 5.46471530125051721767274642188, 6.93652320458930159659085845537, 9.243216947893783679889127414459, 10.01484519749737815149838260535, 12.36024457657215969347376054739, 13.15701193127193847111161585038, 14.47746296938969204241991642606, 15.97186575921780253256672390624
