| L(s) = 1 | + (−0.180 − 0.495i)3-s + (−0.550 + 3.12i)5-s + (−25.9 − 45.0i)7-s + (61.8 − 51.8i)9-s + (−19.1 + 33.1i)11-s + (71.6 − 196. i)13-s + (1.64 − 0.290i)15-s + (−302. − 253. i)17-s + (210. − 293. i)19-s + (−17.6 + 21.0i)21-s + (−80.8 − 458. i)23-s + (577. + 210. i)25-s + (−73.9 − 42.6i)27-s + (277. + 330. i)29-s + (−1.06e3 + 612. i)31-s + ⋯ |
| L(s) = 1 | + (−0.0200 − 0.0550i)3-s + (−0.0220 + 0.124i)5-s + (−0.530 − 0.919i)7-s + (0.763 − 0.640i)9-s + (−0.158 + 0.274i)11-s + (0.424 − 1.16i)13-s + (0.00731 − 0.00129i)15-s + (−1.04 − 0.878i)17-s + (0.582 − 0.812i)19-s + (−0.0399 + 0.0476i)21-s + (−0.152 − 0.867i)23-s + (0.924 + 0.336i)25-s + (−0.101 − 0.0585i)27-s + (0.329 + 0.393i)29-s + (−1.10 + 0.637i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0768 + 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0768 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.00747 - 0.932795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00747 - 0.932795i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-210. + 293. i)T \) |
| good | 3 | \( 1 + (0.180 + 0.495i)T + (-62.0 + 52.0i)T^{2} \) |
| 5 | \( 1 + (0.550 - 3.12i)T + (-587. - 213. i)T^{2} \) |
| 7 | \( 1 + (25.9 + 45.0i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (19.1 - 33.1i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-71.6 + 196. i)T + (-2.18e4 - 1.83e4i)T^{2} \) |
| 17 | \( 1 + (302. + 253. i)T + (1.45e4 + 8.22e4i)T^{2} \) |
| 23 | \( 1 + (80.8 + 458. i)T + (-2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-277. - 330. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (1.06e3 - 612. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 963. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-866. - 2.38e3i)T + (-2.16e6 + 1.81e6i)T^{2} \) |
| 43 | \( 1 + (372. - 2.11e3i)T + (-3.21e6 - 1.16e6i)T^{2} \) |
| 47 | \( 1 + (1.91e3 - 1.60e3i)T + (8.47e5 - 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-3.62e3 + 639. i)T + (7.41e6 - 2.69e6i)T^{2} \) |
| 59 | \( 1 + (897. - 1.06e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (83.2 + 471. i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (1.81e3 + 2.16e3i)T + (-3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (-5.13e3 - 904. i)T + (2.38e7 + 8.69e6i)T^{2} \) |
| 73 | \( 1 + (-3.34e3 + 1.21e3i)T + (2.17e7 - 1.82e7i)T^{2} \) |
| 79 | \( 1 + (2.47e3 + 6.80e3i)T + (-2.98e7 + 2.50e7i)T^{2} \) |
| 83 | \( 1 + (3.49e3 + 6.04e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (194. - 533. i)T + (-4.80e7 - 4.03e7i)T^{2} \) |
| 97 | \( 1 + (-6.98e3 + 8.31e3i)T + (-1.53e7 - 8.71e7i)T^{2} \) |
| 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
−13.25741723987024836891344367847, −12.73367725950078035923827182807, −11.14665279032123229216059578342, −10.19639500429133390247880220715, −9.078142594356920328317809385105, −7.42359476809989819189713848594, −6.56917626251312896879664422652, −4.70528047492773882971647195947, −3.17122647022692493102566082001, −0.72915097438904069984910491428,
1.98173130641762640288472443372, 3.97625533804671254809057981914, 5.58745175682674030641394234534, 6.89242021648253767760137624498, 8.429312870192656034085037786785, 9.452023583285921463182636545639, 10.70440357171852156206837845783, 11.90449379166674628804622621783, 12.95714864445092828672925900725, 13.87864936118685776971996702691
