| L(s) = 1 | + (12.6 − 8.15i)2-s + (−2.21 + 15.4i)3-s + (67.9 − 148. i)4-s + (−11.0 − 37.7i)5-s + (97.6 + 213. i)6-s + (105. − 91.7i)7-s + (−213. − 1.48e3i)8-s + (−233. − 68.4i)9-s + (−448. − 389. i)10-s + (684. − 1.06e3i)11-s + (2.14e3 + 1.37e3i)12-s + (281. − 325. i)13-s + (595. − 2.02e3i)14-s + (607. − 87.3i)15-s + (−7.96e3 − 9.19e3i)16-s + (−2.15e3 + 986. i)17-s + ⋯ |
| L(s) = 1 | + (1.58 − 1.01i)2-s + (−0.0821 + 0.571i)3-s + (1.06 − 2.32i)4-s + (−0.0887 − 0.302i)5-s + (0.452 + 0.990i)6-s + (0.308 − 0.267i)7-s + (−0.417 − 2.90i)8-s + (−0.319 − 0.0939i)9-s + (−0.448 − 0.389i)10-s + (0.514 − 0.800i)11-s + (1.24 + 0.797i)12-s + (0.128 − 0.148i)13-s + (0.216 − 0.738i)14-s + (0.180 − 0.0258i)15-s + (−1.94 − 2.24i)16-s + (−0.439 + 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.42594 - 3.59421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.42594 - 3.59421i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.21 - 15.4i)T \) |
| 23 | \( 1 + (-1.00e4 - 6.78e3i)T \) |
| good | 2 | \( 1 + (-12.6 + 8.15i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (11.0 + 37.7i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (-105. + 91.7i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (-684. + 1.06e3i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (-281. + 325. i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (2.15e3 - 986. i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (2.91e3 + 1.33e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (-3.03e3 - 6.65e3i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (-2.79e3 - 1.94e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (2.07e4 - 7.06e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (9.43e4 - 2.77e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (-1.14e5 - 1.64e4i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 1.06e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.07e5 + 9.33e4i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (2.27e5 - 2.62e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (-3.39e5 + 4.88e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (8.31e4 + 1.29e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (-3.96e5 + 2.54e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (1.50e5 - 3.30e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (5.17e5 + 4.48e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (-8.47e4 + 2.88e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (3.06e5 + 4.40e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (1.37e5 + 4.69e5i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.19399840364050734000094637252, −12.06121242819504461687693477862, −11.13045227450255443900933124601, −10.38497733522316655026067148400, −8.858823939284672155730071758012, −6.52276092218143740776460400175, −5.21600635497560025570679388723, −4.21396264727411651847067081052, −3.01375732186860094568739780797, −1.14451317939203808034509293377,
2.38529289269200764179588437499, 4.05475492833515963831463895319, 5.33921493559682341229877048776, 6.62859140759583114246822095833, 7.33908986820930095800059558778, 8.718844451282566087428936966160, 11.09699499344103744752281851057, 12.16805280233407502447700838845, 12.89939100311634495718979540168, 14.00296761023567033227161410551
