| L(s) = 1 | + (−1.18 + 1.87i)2-s + (2.12 − 1.99i)3-s + (−1.24 − 2.64i)4-s + (−1.16 − 1.91i)5-s + (1.21 + 6.36i)6-s + (1.89 − 1.37i)7-s + (2.03 + 0.257i)8-s + (0.346 − 5.50i)9-s + (4.96 + 0.0937i)10-s + (0.422 − 0.665i)11-s + (−7.93 − 3.14i)12-s + (−0.320 + 5.09i)13-s + (0.325 + 5.17i)14-s + (−6.28 − 1.74i)15-s + (0.825 − 0.998i)16-s + (−2.92 + 6.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.840 + 1.32i)2-s + (1.22 − 1.15i)3-s + (−0.622 − 1.32i)4-s + (−0.519 − 0.854i)5-s + (0.495 + 2.59i)6-s + (0.715 − 0.519i)7-s + (0.719 + 0.0909i)8-s + (0.115 − 1.83i)9-s + (1.56 + 0.0296i)10-s + (0.127 − 0.200i)11-s + (−2.29 − 0.907i)12-s + (−0.0888 + 1.41i)13-s + (0.0870 + 1.38i)14-s + (−1.62 − 0.449i)15-s + (0.206 − 0.249i)16-s + (−0.708 + 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.957509 + 0.00896879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.957509 + 0.00896879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.16 + 1.91i)T \) |
| good | 2 | \( 1 + (1.18 - 1.87i)T + (-0.851 - 1.80i)T^{2} \) |
| 3 | \( 1 + (-2.12 + 1.99i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (-1.89 + 1.37i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.422 + 0.665i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (0.320 - 5.09i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (2.92 - 6.20i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (-2.64 - 2.48i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.88 - 0.484i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (3.82 + 2.10i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (-2.80 + 5.96i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (-2.98 + 3.61i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (-5.09 + 1.30i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (0.747 - 2.29i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (5.44 - 0.688i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (0.0246 - 0.129i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (-7.51 - 2.97i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (14.2 + 3.65i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (0.115 - 0.0632i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (8.69 - 1.09i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (1.73 - 0.686i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (6.32 - 5.93i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-4.09 - 3.84i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-4.95 + 1.96i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (7.75 + 4.26i)T + (51.9 + 81.8i)T^{2} \) |
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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
−13.70510366188818329784913755927, −12.69844182968214919159569777227, −11.46314588433217802744816798848, −9.446615129262698543668729731094, −8.704567420795891164888721110989, −7.945599462821666075001681096873, −7.35903128546566249570714079562, −6.16288973044430937262777204805, −4.16555843397059353846829873828, −1.50392120234846137196128723172,
2.62085430713756077674201736976, 3.25012533690672250049705573309, 4.82021694529700680384130208730, 7.56154258341874117033991122425, 8.529064724750639461864532226105, 9.351619714178036768692238309787, 10.26240111955052570547409771262, 11.03866819002013830070644601252, 11.85849254985159460221540090585, 13.36449206659306573016800431131
