| L(s) = 1 | + (−13.0 + 9.44i)2-s + (−6.54 + 20.1i)3-s + (40.2 − 124. i)4-s + (−242. + 138. i)5-s + (−105. − 323. i)6-s − 1.23e3·7-s + (11.9 + 36.7i)8-s + (1.40e3 + 1.02e3i)9-s + (1.85e3 − 4.09e3i)10-s + (2.54e3 − 1.84e3i)11-s + (2.23e3 + 1.62e3i)12-s + (5.80e3 + 4.21e3i)13-s + (1.60e4 − 1.16e4i)14-s + (−1.19e3 − 5.79e3i)15-s + (1.30e4 + 9.44e3i)16-s + (−1.14e4 − 3.52e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 + 0.835i)2-s + (−0.139 + 0.430i)3-s + (0.314 − 0.968i)4-s + (−0.868 + 0.494i)5-s + (−0.198 − 0.612i)6-s − 1.35·7-s + (0.00825 + 0.0254i)8-s + (0.642 + 0.467i)9-s + (0.585 − 1.29i)10-s + (0.575 − 0.418i)11-s + (0.373 + 0.271i)12-s + (0.732 + 0.531i)13-s + (1.55 − 1.13i)14-s + (−0.0915 − 0.443i)15-s + (0.793 + 0.576i)16-s + (−0.565 − 1.74i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.198053 - 0.0749786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.198053 - 0.0749786i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (242. - 138. i)T \) |
| good | 2 | \( 1 + (13.0 - 9.44i)T + (39.5 - 121. i)T^{2} \) |
| 3 | \( 1 + (6.54 - 20.1i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 7 | \( 1 + 1.23e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + (-2.54e3 + 1.84e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-5.80e3 - 4.21e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (1.14e4 + 3.52e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-3.90e3 - 1.20e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (3.31e4 - 2.40e4i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-1.33e4 + 4.10e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (7.96e4 + 2.45e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.09e5 - 1.52e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (5.91e5 + 4.30e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 2.12e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.85e5 + 5.72e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (4.86e4 - 1.49e5i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-8.11e4 - 5.89e4i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (1.24e6 - 9.07e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-3.50e5 - 1.07e6i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (2.21e4 - 6.82e4i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-3.00e6 + 2.18e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-9.25e5 + 2.84e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (2.20e6 + 6.79e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (-6.12e6 + 4.44e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-4.53e6 + 1.39e7i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−16.09661165443504964230567487355, −15.38165368257574796624241014886, −13.49684858155581334186630194077, −11.62375524488946415963295865152, −10.10840097744037716515914480682, −9.094884156493262100088653356586, −7.47067984780499897665175602030, −6.43046775281158305025497184228, −3.74712710072885717571872613319, −0.18621531017313901668980327504,
1.25912142835854836849024358369, 3.63399234196183111466996601793, 6.59861207532933901439002333146, 8.301256297038656223874400778298, 9.481042447961851489221796198240, 10.75608400581592421768219254061, 12.25992276290224187201018946250, 12.87403117738930006806559575966, 15.26122750569903852707641356220, 16.42961564206332939237611561401
