| L(s) = 1 | + (12.6 − 23.8i)3-s + (152. − 88.2i)5-s + (152. − 264. i)7-s + (−409. − 602. i)9-s + (458. + 264. i)11-s + (1.60e3 + 2.78e3i)13-s + (−174. − 4.76e3i)15-s − 2.53e3i·17-s − 9.07e3·19-s + (−4.37e3 − 6.97e3i)21-s + (9.31e3 − 5.37e3i)23-s + (7.76e3 − 1.34e4i)25-s + (−1.95e4 + 2.16e3i)27-s + (−559. − 323. i)29-s + (−1.53e4 − 2.65e4i)31-s + ⋯ |
| L(s) = 1 | + (0.467 − 0.883i)3-s + (1.22 − 0.705i)5-s + (0.444 − 0.770i)7-s + (−0.562 − 0.826i)9-s + (0.344 + 0.198i)11-s + (0.731 + 1.26i)13-s + (−0.0518 − 1.41i)15-s − 0.516i·17-s − 1.32·19-s + (−0.472 − 0.753i)21-s + (0.765 − 0.441i)23-s + (0.496 − 0.860i)25-s + (−0.993 + 0.109i)27-s + (−0.0229 − 0.0132i)29-s + (−0.515 − 0.892i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.82403 - 2.01838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82403 - 2.01838i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-12.6 + 23.8i)T \) |
| good | 5 | \( 1 + (-152. + 88.2i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-152. + 264. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-458. - 264. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.60e3 - 2.78e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 2.53e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 9.07e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-9.31e3 + 5.37e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (559. + 323. i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.53e4 + 2.65e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 3.25e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (1.13e5 - 6.54e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-7.61e4 + 1.31e5i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-8.24e4 - 4.76e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.53e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (1.27e5 - 7.38e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.46e5 - 2.54e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.08e5 - 1.87e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 2.32e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.92e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-3.79e5 + 6.57e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.54e5 - 2.04e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 1.05e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-2.09e5 + 3.62e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
| 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
−13.42895150951036800899017349800, −12.28209947571630734321192434190, −10.94076286149298546726024863730, −9.384594455862929685845185858604, −8.641118300998204922965038909626, −7.11163853199456507238669747927, −6.04922141697886546897515683838, −4.31106074543992505892815235100, −2.10957526205784005769978443384, −1.08246492708917647651657477076,
2.04369363545398686556403935967, 3.34434328713602565765514849425, 5.23756879007450131106636703859, 6.24735774251049594971966366528, 8.255132625751425676107376154125, 9.192726985649713144897068606259, 10.40060011299281828535303848473, 11.03040744671352765996593064615, 12.84701528696137895214064148286, 13.91905709166698808266723789552
