| L(s) = 1 | + (−1.56e4 + 9.04e3i)2-s + (−8.12e6 − 1.63e6i)3-s + (−1.04e8 + 1.81e8i)4-s + (5.75e9 + 9.96e9i)5-s + (1.42e11 − 4.77e10i)6-s + (6.04e11 + 1.68e12i)7-s + (0.000488 − 1.35e13i)8-s + (6.32e13 + 2.66e13i)9-s + (−1.80e14 − 1.04e14i)10-s + (7.97e13 + 4.60e13i)11-s + (1.14e15 − 1.30e15i)12-s + 6.58e15i·13-s + (−2.47e16 − 2.10e16i)14-s + (−3.03e16 − 9.03e16i)15-s + (6.59e16 + 1.14e17i)16-s + (3.20e17 − 5.55e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.676 + 0.390i)2-s + (−0.980 − 0.197i)3-s + (−0.195 + 0.337i)4-s + (0.421 + 0.730i)5-s + (0.740 − 0.248i)6-s + (0.336 + 0.941i)7-s − 1.08i·8-s + (0.921 + 0.387i)9-s + (−0.570 − 0.329i)10-s + (0.0632 + 0.0365i)11-s + (0.258 − 0.292i)12-s + 0.463i·13-s + (−0.595 − 0.505i)14-s + (−0.268 − 0.799i)15-s + (0.228 + 0.396i)16-s + (0.461 − 0.799i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.5852976422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5852976422\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.12e6 + 1.63e6i)T \) |
| 7 | \( 1 + (-6.04e11 - 1.68e12i)T \) |
| good | 2 | \( 1 + (1.56e4 - 9.04e3i)T + (2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-5.75e9 - 9.96e9i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-7.97e13 - 4.60e13i)T + (7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 - 6.58e15iT - 2.01e32T^{2} \) |
| 17 | \( 1 + (-3.20e17 + 5.55e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (2.57e18 - 1.48e18i)T + (6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (3.71e19 - 2.14e19i)T + (1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 - 6.08e20iT - 2.56e42T^{2} \) |
| 31 | \( 1 + (-1.38e21 - 8.01e20i)T + (8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-2.16e22 - 3.74e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 7.14e22T + 5.89e46T^{2} \) |
| 43 | \( 1 - 4.32e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-1.10e24 - 1.90e24i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-2.93e24 - 1.69e24i)T + (5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (9.67e23 - 1.67e24i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (1.18e26 - 6.82e25i)T + (2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (6.61e25 - 1.14e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 - 1.14e27iT - 4.85e53T^{2} \) |
| 73 | \( 1 + (-8.27e26 - 4.77e26i)T + (5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-5.93e26 - 1.02e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 - 2.72e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (9.15e27 + 1.58e28i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 + 1.91e28iT - 4.13e57T^{2} \) |
| 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
−12.62789714152067116094250883261, −11.70762970738086515094356729123, −10.35928135397266580569914587764, −9.256495379938751566182193790274, −7.87458488234780623463991434695, −6.74366698327098228088923070087, −5.79790977246891931222886691548, −4.34466530627142039084551536288, −2.61603428571264152705634824856, −1.21916070877613898957179422843,
0.24408759298063652829737978682, 0.911224151780741100383830928016, 1.86313948220835734923449889398, 4.13283672832054147832057156084, 5.13142670574873024846872651555, 6.17775929481258074672713459589, 7.890549531547156033376971019833, 9.250709858957552521080990913074, 10.33536424433220480136612013656, 10.95543844205767122281673804194
