L(s) = 1 | − 4.76e9i·2-s + (4.09e15 − 3.75e15i)3-s + 5.11e19·4-s + 1.11e23i·5-s + (−1.78e25 − 1.95e25i)6-s + 1.00e28·7-s − 5.94e29i·8-s + (2.66e30 − 3.07e31i)9-s + 5.31e32·10-s − 2.71e34i·11-s + (2.09e35 − 1.92e35i)12-s − 8.16e36·13-s − 4.77e37i·14-s + (4.19e38 + 4.57e38i)15-s + 9.38e38·16-s + 3.36e40i·17-s + ⋯ |
L(s) = 1 | − 0.554i·2-s + (0.736 − 0.675i)3-s + 0.692·4-s + 0.958i·5-s + (−0.374 − 0.408i)6-s + 1.29·7-s − 0.938i·8-s + (0.0862 − 0.996i)9-s + 0.531·10-s − 1.16i·11-s + (0.510 − 0.468i)12-s − 1.41·13-s − 0.719i·14-s + (0.648 + 0.706i)15-s + 0.172·16-s + 0.835i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{67}{2})\) |
\(\approx\) |
\(3.435210110\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.435210110\) |
\(L(34)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.09e15 + 3.75e15i)T \) |
good | 2 | \( 1 + 4.76e9iT - 7.37e19T^{2} \) |
| 5 | \( 1 - 1.11e23iT - 1.35e46T^{2} \) |
| 7 | \( 1 - 1.00e28T + 5.97e55T^{2} \) |
| 11 | \( 1 + 2.71e34iT - 5.39e68T^{2} \) |
| 13 | \( 1 + 8.16e36T + 3.31e73T^{2} \) |
| 17 | \( 1 - 3.36e40iT - 1.62e81T^{2} \) |
| 19 | \( 1 + 2.59e42T + 2.49e84T^{2} \) |
| 23 | \( 1 + 7.82e44iT - 7.48e89T^{2} \) |
| 29 | \( 1 + 2.09e48iT - 3.29e96T^{2} \) |
| 31 | \( 1 + 9.44e48T + 2.69e98T^{2} \) |
| 37 | \( 1 - 5.25e51T + 3.17e103T^{2} \) |
| 41 | \( 1 + 2.65e53iT - 2.77e106T^{2} \) |
| 43 | \( 1 - 7.04e53T + 6.44e107T^{2} \) |
| 47 | \( 1 + 5.54e54iT - 2.28e110T^{2} \) |
| 53 | \( 1 + 5.01e56iT - 6.34e113T^{2} \) |
| 59 | \( 1 - 2.39e58iT - 7.52e116T^{2} \) |
| 61 | \( 1 - 5.18e58T + 6.78e117T^{2} \) |
| 67 | \( 1 - 1.99e60T + 3.31e120T^{2} \) |
| 71 | \( 1 - 1.53e61iT - 1.52e122T^{2} \) |
| 73 | \( 1 + 8.33e60T + 9.53e122T^{2} \) |
| 79 | \( 1 - 2.10e62T + 1.75e125T^{2} \) |
| 83 | \( 1 + 2.23e63iT - 4.56e126T^{2} \) |
| 89 | \( 1 - 2.08e62iT - 4.56e128T^{2} \) |
| 97 | \( 1 - 1.02e65T + 1.33e131T^{2} \) |
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
−12.67167013656219702127051547730, −11.38709559311669810414537437929, −10.41752871918042685150139518772, −8.440080206873965166104675118689, −7.33173206897391267251110176182, −6.19527024075224086376459487723, −3.95071224110137476033030890303, −2.52310943402345800952601742967, −2.08256567985446905513728970631, −0.61212563668496490303518114178,
1.62691977392056023388442773195, 2.46285504623076245985277787830, 4.58687488381421422839228257199, 5.08229920374033864776675817286, 7.29107439437943214101323618928, 8.172509053211805990335381008060, 9.478630033443232278544098251595, 11.04354547012486545680721336511, 12.52113621171521792685871445470, 14.51731919321309968222629950116