L(s) = 1 | + (−221. + 221. i)3-s + (−7.91e3 − 7.91e3i)5-s + 7.68e4i·7-s + 7.90e4i·9-s + (1.97e5 + 1.97e5i)11-s + (−4.68e4 + 4.68e4i)13-s + 3.50e6·15-s + 9.74e6·17-s + (−2.85e6 + 2.85e6i)19-s + (−1.70e7 − 1.70e7i)21-s + 3.12e7i·23-s + 7.65e7i·25-s + (−5.67e7 − 5.67e7i)27-s + (−1.26e8 + 1.26e8i)29-s − 3.46e7·31-s + ⋯ |
L(s) = 1 | + (−0.526 + 0.526i)3-s + (−1.13 − 1.13i)5-s + 1.72i·7-s + 0.446i·9-s + (0.370 + 0.370i)11-s + (−0.0350 + 0.0350i)13-s + 1.19·15-s + 1.66·17-s + (−0.264 + 0.264i)19-s + (−0.909 − 0.909i)21-s + 1.01i·23-s + 1.56i·25-s + (−0.761 − 0.761i)27-s + (−1.14 + 1.14i)29-s − 0.217·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.641i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.113240 - 0.312191i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.113240 - 0.312191i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (221. - 221. i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (7.91e3 + 7.91e3i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 - 7.68e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (-1.97e5 - 1.97e5i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (4.68e4 - 4.68e4i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 - 9.74e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (2.85e6 - 2.85e6i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 - 3.12e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (1.26e8 - 1.26e8i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 + 3.46e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-2.25e8 - 2.25e8i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 + 4.55e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (7.03e8 + 7.03e8i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 + 6.32e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + (1.37e9 + 1.37e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (1.10e9 + 1.10e9i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (-3.14e9 + 3.14e9i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (-3.21e9 + 3.21e9i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 - 1.72e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 1.53e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 4.44e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + (-1.61e10 + 1.61e10i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 - 1.60e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 7.16e9T + 7.15e21T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.80710348254314467495377713668, −12.05163859046542652783305480800, −11.40256637803294239318308129087, −9.733951848668619509849631999097, −8.695017660913345951372968356655, −7.70760059815779926402363992632, −5.60464716567871934560705578960, −4.99305783815984578301573893453, −3.56384328238197579905490416340, −1.63809977725701834766844202128,
0.11997579760025520009742456477, 0.998814486213607827903314559404, 3.29234848135880362508366347392, 4.11848081308364456990650959649, 6.24010738694155303038832710408, 7.18731701364488810564668746076, 7.85973201731410839242919570693, 9.958471671020102481380811562543, 11.00860162134614668212603114559, 11.68977917306308109696043256157