| L(s) = 1 | + (−2.41 − 3.31i)2-s + (−18.6 − 6.04i)3-s + (4.68 − 14.4i)4-s + (9.93 + 55.0i)5-s + (24.8 + 76.3i)6-s + 138. i·7-s + (−184. + 59.8i)8-s + (112. + 82.0i)9-s + (158. − 165. i)10-s + (−21.2 + 15.4i)11-s + (−174. + 239. i)12-s + (−108. + 149. i)13-s + (458. − 333. i)14-s + (147. − 1.08e3i)15-s + (249. + 181. i)16-s + (−1.66e3 + 539. i)17-s + ⋯ |
| L(s) = 1 | + (−0.426 − 0.586i)2-s + (−1.19 − 0.387i)3-s + (0.146 − 0.450i)4-s + (0.177 + 0.984i)5-s + (0.281 + 0.865i)6-s + 1.06i·7-s + (−1.01 + 0.330i)8-s + (0.465 + 0.337i)9-s + (0.501 − 0.523i)10-s + (−0.0528 + 0.0383i)11-s + (−0.349 + 0.481i)12-s + (−0.178 + 0.245i)13-s + (0.625 − 0.454i)14-s + (0.169 − 1.24i)15-s + (0.243 + 0.177i)16-s + (−1.39 + 0.452i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.164188 + 0.194534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.164188 + 0.194534i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-9.93 - 55.0i)T \) |
| good | 2 | \( 1 + (2.41 + 3.31i)T + (-9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (18.6 + 6.04i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 - 138. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (21.2 - 15.4i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (108. - 149. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.66e3 - 539. i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (84.9 + 261. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (988. + 1.36e3i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (2.43e3 - 7.50e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (2.14e3 + 6.61e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.58e3 + 2.17e3i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.05e4 - 7.68e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 2.39e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.31e4 - 4.28e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (3.68e4 + 1.19e4i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.77e4 + 1.28e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.25e4 - 9.09e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-5.96e3 + 1.93e3i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.38e4 - 4.26e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-3.89e4 - 5.35e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.11e4 - 6.50e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-4.52e4 + 1.46e4i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-2.51e4 + 1.82e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-6.70e4 - 2.17e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−17.37531654552183921725124966707, −15.61505077511885476843505012013, −14.55612947864817142368163601285, −12.55152275855761710234869646080, −11.38171758431934393799390132501, −10.74010091644850971848928513459, −9.157074018335457348169534524917, −6.65659302981272381480240002738, −5.67346341476556254273086331580, −2.22393206258353519190862542223,
0.21453598630182444204221944990, 4.46978392146202966980932266364, 6.15791360156494095620273096638, 7.75068608309270228198388347554, 9.385228300934455450949923182983, 10.96713870822293025923913056740, 12.20378434289501350544250300645, 13.49696407954205261299042258495, 15.68750443514397808079522221698, 16.42787348025684964285873420881
