| L(s) = 1 | + (4.90e5 − 2.03e5i)3-s − 1.24e8i·5-s − 1.50e10·7-s + (1.99e11 − 2.00e11i)9-s − 2.25e12i·11-s + 4.78e12·13-s + (−2.52e13 − 6.08e13i)15-s − 5.31e14i·17-s + 2.04e15·19-s + (−7.39e15 + 3.06e15i)21-s − 3.16e16i·23-s + 4.42e16·25-s + (5.71e16 − 1.38e17i)27-s + 6.82e17i·29-s + 6.83e17·31-s + ⋯ |
| L(s) = 1 | + (0.923 − 0.383i)3-s − 0.507i·5-s − 1.08·7-s + (0.705 − 0.708i)9-s − 0.719i·11-s + 0.205·13-s + (−0.194 − 0.469i)15-s − 0.912i·17-s + 0.924·19-s + (−1.00 + 0.417i)21-s − 1.44i·23-s + 0.741·25-s + (0.380 − 0.924i)27-s + 1.92i·29-s + 0.867·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{25}{2})\) |
\(\approx\) |
\(2.322061966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.322061966\) |
| \(L(13)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.90e5 + 2.03e5i)T \) |
| good | 5 | \( 1 + 1.24e8iT - 5.96e16T^{2} \) |
| 7 | \( 1 + 1.50e10T + 1.91e20T^{2} \) |
| 11 | \( 1 + 2.25e12iT - 9.84e24T^{2} \) |
| 13 | \( 1 - 4.78e12T + 5.42e26T^{2} \) |
| 17 | \( 1 + 5.31e14iT - 3.39e29T^{2} \) |
| 19 | \( 1 - 2.04e15T + 4.89e30T^{2} \) |
| 23 | \( 1 + 3.16e16iT - 4.80e32T^{2} \) |
| 29 | \( 1 - 6.82e17iT - 1.25e35T^{2} \) |
| 31 | \( 1 - 6.83e17T + 6.20e35T^{2} \) |
| 37 | \( 1 - 9.74e18T + 4.33e37T^{2} \) |
| 41 | \( 1 + 1.91e19iT - 5.09e38T^{2} \) |
| 43 | \( 1 + 6.07e19T + 1.59e39T^{2} \) |
| 47 | \( 1 + 1.14e20iT - 1.35e40T^{2} \) |
| 53 | \( 1 - 5.21e20iT - 2.41e41T^{2} \) |
| 59 | \( 1 + 5.99e20iT - 3.16e42T^{2} \) |
| 61 | \( 1 - 7.42e19T + 7.04e42T^{2} \) |
| 67 | \( 1 + 8.52e21T + 6.69e43T^{2} \) |
| 71 | \( 1 + 2.05e22iT - 2.69e44T^{2} \) |
| 73 | \( 1 + 3.58e22T + 5.24e44T^{2} \) |
| 79 | \( 1 + 2.61e22T + 3.49e45T^{2} \) |
| 83 | \( 1 + 1.17e23iT - 1.14e46T^{2} \) |
| 89 | \( 1 + 1.60e23iT - 6.10e46T^{2} \) |
| 97 | \( 1 + 8.32e22T + 4.81e47T^{2} \) |
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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
−10.34293112562444069332788271031, −9.208581428909895941450161308913, −8.558426566617763007507567595305, −7.22988683268104517517386985023, −6.28636451059349168168839712552, −4.78865012192141584667241809036, −3.35214261794237376383883971292, −2.76263738145255698869376554296, −1.21387857195471855848124061284, −0.38964081166790271834485793612,
1.34389862486389762774753965612, 2.63773778886861712571719050481, 3.39398271351186455444275739897, 4.44527536833935794770569213412, 6.04248170033128726278061417406, 7.18783484095540620762314073798, 8.208535717938032579400314054187, 9.676464026697590901758573034377, 9.945815502999235821762274085637, 11.45585512582938517553815098117
