L(s) = 1 | + (−18.1 + 18.1i)2-s + (95.3 + 95.3i)3-s − 401. i·4-s − 3.45e3·6-s + (−1.47e3 + 1.47e3i)7-s + (2.63e3 + 2.63e3i)8-s + 1.16e4i·9-s − 1.63e4·11-s + (3.82e4 − 3.82e4i)12-s + (−1.17e4 − 1.17e4i)13-s − 5.36e4i·14-s + 7.29e3·16-s + (−2.01e4 + 2.01e4i)17-s + (−2.10e5 − 2.10e5i)18-s − 1.19e5i·19-s + ⋯ |
L(s) = 1 | + (−1.13 + 1.13i)2-s + (1.17 + 1.17i)3-s − 1.56i·4-s − 2.66·6-s + (−0.615 + 0.615i)7-s + (0.642 + 0.642i)8-s + 1.77i·9-s − 1.11·11-s + (1.84 − 1.84i)12-s + (−0.410 − 0.410i)13-s − 1.39i·14-s + 0.111·16-s + (−0.241 + 0.241i)17-s + (−2.00 − 2.00i)18-s − 0.920i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.384871 - 0.615314i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.384871 - 0.615314i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (18.1 - 18.1i)T - 256iT^{2} \) |
| 3 | \( 1 + (-95.3 - 95.3i)T + 6.56e3iT^{2} \) |
| 7 | \( 1 + (1.47e3 - 1.47e3i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + 1.63e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (1.17e4 + 1.17e4i)T + 8.15e8iT^{2} \) |
| 17 | \( 1 + (2.01e4 - 2.01e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + 1.19e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-3.61e5 - 3.61e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 - 6.49e4iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 9.06e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (1.51e6 - 1.51e6i)T - 3.51e12iT^{2} \) |
| 41 | \( 1 - 2.57e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (2.04e6 + 2.04e6i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + (1.34e6 - 1.34e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 + (-7.73e5 - 7.73e5i)T + 6.22e13iT^{2} \) |
| 59 | \( 1 - 1.56e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 8.14e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + (2.76e7 - 2.76e7i)T - 4.06e14iT^{2} \) |
| 71 | \( 1 + 1.45e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.43e7 - 1.43e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + 4.59e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-2.00e7 - 2.00e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 - 3.53e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (5.33e6 - 5.33e6i)T - 7.83e15iT^{2} \) |
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
−16.20803833120455275183164635360, −15.45418907560443191966378927339, −14.93722628799771173141717452593, −13.23435890472294306200061158591, −10.52098551876014043923066811300, −9.445943882599754691582681779206, −8.702351913098314113474345132019, −7.40562347216588560971770649071, −5.31429318708595659938293643241, −2.96290071386859504122507825078,
0.40526502697941201731541400554, 1.99848707957587854284374841971, 3.17009986138428465639071615458, 7.11789601561456663449540353477, 8.219868523968653314570313104510, 9.384907992020503100638775228053, 10.68427231865724774107697416607, 12.42676287147103031011640912547, 13.16579280389356799138188558208, 14.55767592805921344632119517161