L(s) = 1 | + (4.39 + 4.39i)2-s + (−75.2 + 75.2i)3-s − 217. i·4-s − 662.·6-s + (−730. − 730. i)7-s + (2.08e3 − 2.08e3i)8-s − 4.77e3i·9-s + 1.95e4·11-s + (1.63e4 + 1.63e4i)12-s + (2.49e4 − 2.49e4i)13-s − 6.42e3i·14-s − 3.73e4·16-s + (1.12e4 + 1.12e4i)17-s + (2.10e4 − 2.10e4i)18-s − 1.71e5i·19-s + ⋯ |
L(s) = 1 | + (0.274 + 0.274i)2-s + (−0.929 + 0.929i)3-s − 0.849i·4-s − 0.510·6-s + (−0.304 − 0.304i)7-s + (0.508 − 0.508i)8-s − 0.728i·9-s + 1.33·11-s + (0.789 + 0.789i)12-s + (0.872 − 0.872i)13-s − 0.167i·14-s − 0.569·16-s + (0.135 + 0.135i)17-s + (0.200 − 0.200i)18-s − 1.31i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.33379 - 0.378904i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33379 - 0.378904i\) |
\(L(5)\) |
|
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 | \( 1 + (-4.39 - 4.39i)T + 256iT^{2} \) |
| 3 | \( 1 + (75.2 - 75.2i)T - 6.56e3iT^{2} \) |
| 7 | \( 1 + (730. + 730. i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 - 1.95e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (-2.49e4 + 2.49e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (-1.12e4 - 1.12e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 + 1.71e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.32e5 + 1.32e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + 1.27e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 9.60e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (-2.43e5 - 2.43e5i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 - 2.50e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (-6.76e3 + 6.76e3i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (-1.79e6 - 1.79e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + (-2.97e6 + 2.97e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 - 3.13e5iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.76e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (4.41e6 + 4.41e6i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 + 8.89e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (1.95e7 - 1.95e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 1.11e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (1.58e7 - 1.58e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 - 4.85e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-1.07e8 - 1.07e8i)T + 7.83e15iT^{2} \) |
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
−15.69188106553845680402700726898, −14.66321581147272136325234620407, −13.23037904186019588483393518567, −11.34522605248965603547364656575, −10.49139343830622792779583102586, −9.256476683093154628116969632934, −6.65441055594003653837770060219, −5.47467305769941621393803982258, −4.09557890360841922166127195116, −0.75043457737565658007698792307,
1.51791422677900550244270476584, 3.81200992741517452995036326414, 6.02639631186924689094996340606, 7.24469024020373587475844577141, 8.985335390637238358964973044113, 11.28266774480938344858038193724, 12.02504232126976264003380163469, 12.92574316951554297403222941477, 14.22597322679657479461642223998, 16.32629322895145909505603436191