| L(s) = 1 | + (−7.18 + 4.15i)2-s + (15.2 + 3.15i)3-s + (18.4 − 31.9i)4-s − 13.4·5-s + (−122. + 40.6i)6-s + (121. + 44.9i)7-s + 40.6i·8-s + (223. + 96.4i)9-s + (96.8 − 55.9i)10-s − 605. i·11-s + (382. − 429. i)12-s + (362. − 209. i)13-s + (−1.06e3 + 181. i)14-s + (−205. − 42.5i)15-s + (421. + 730. i)16-s + (165. + 286. i)17-s + ⋯ |
| L(s) = 1 | + (−1.27 + 0.733i)2-s + (0.979 + 0.202i)3-s + (0.576 − 0.998i)4-s − 0.240·5-s + (−1.39 + 0.460i)6-s + (0.937 + 0.346i)7-s + 0.224i·8-s + (0.917 + 0.396i)9-s + (0.306 − 0.176i)10-s − 1.50i·11-s + (0.766 − 0.860i)12-s + (0.595 − 0.343i)13-s + (−1.44 + 0.247i)14-s + (−0.235 − 0.0488i)15-s + (0.411 + 0.713i)16-s + (0.138 + 0.240i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.20762 + 0.684154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20762 + 0.684154i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-15.2 - 3.15i)T \) |
| 7 | \( 1 + (-121. - 44.9i)T \) |
| good | 2 | \( 1 + (7.18 - 4.15i)T + (16 - 27.7i)T^{2} \) |
| 5 | \( 1 + 13.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 605. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-362. + 209. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-165. - 286. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.72e3 - 994. i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 - 3.11e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (83.1 + 48.0i)T + (1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-4.15e3 - 2.39e3i)T + (1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-58.2 + 100. i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-7.29e3 - 1.26e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.01e4 + 1.75e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (6.87e3 + 1.19e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.92e3 - 1.11e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-8.37e3 + 1.45e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (8.98e3 - 5.18e3i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.33e4 + 2.31e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 1.31e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (6.31e4 - 3.64e4i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (4.74e4 + 8.22e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-4.64e3 + 8.04e3i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-4.91e4 + 8.51e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (4.09e4 + 2.36e4i)T + (4.29e9 + 7.43e9i)T^{2} \) |
| 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
−14.40480301356505119340867866277, −13.43433827099200338286684064218, −11.56031325390121570152738564190, −10.30306426167526945822526764304, −9.077767586934087416848185864435, −8.232647494318825030650066225827, −7.62549648411774724521766233419, −5.76883753852625760868392876096, −3.53825040645110082993802636219, −1.26827428340905619767487720689,
1.18899151060180133593055755921, 2.43942100633965053951614130351, 4.41031556329773893088335692330, 7.27888077922718864196319137032, 8.044886341270328803091147611471, 9.191423024714610776555037810718, 10.06787720556327125132169209814, 11.31587246061447743407309121865, 12.36907431563629178163852037224, 13.86360830718444067859385308746
