| L(s) = 1 | + (−15.5 − 3.63i)2-s + (34.7 + 83.9i)3-s + (229. + 113. i)4-s + (212. − 514. i)5-s + (−236. − 1.43e3i)6-s + (423. − 423. i)7-s + (−3.16e3 − 2.60e3i)8-s + (−1.19e3 + 1.19e3i)9-s + (−5.18e3 + 7.23e3i)10-s + (170. − 412. i)11-s + (−1.53e3 + 2.32e4i)12-s + (−1.11e3 − 2.68e3i)13-s + (−8.13e3 + 5.05e3i)14-s + 5.05e4·15-s + (3.98e4 + 5.20e4i)16-s − 1.01e5i·17-s + ⋯ |
| L(s) = 1 | + (−0.973 − 0.227i)2-s + (0.429 + 1.03i)3-s + (0.896 + 0.442i)4-s + (0.340 − 0.822i)5-s + (−0.182 − 1.10i)6-s + (0.176 − 0.176i)7-s + (−0.772 − 0.635i)8-s + (−0.181 + 0.181i)9-s + (−0.518 + 0.723i)10-s + (0.0116 − 0.0281i)11-s + (−0.0739 + 1.11i)12-s + (−0.0389 − 0.0941i)13-s + (−0.211 + 0.131i)14-s + 0.998·15-s + (0.607 + 0.794i)16-s − 1.21i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.50138 - 0.124209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50138 - 0.124209i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.5 + 3.63i)T \) |
| good | 3 | \( 1 + (-34.7 - 83.9i)T + (-4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (-212. + 514. i)T + (-2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-423. + 423. i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + (-170. + 412. i)T + (-1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (1.11e3 + 2.68e3i)T + (-5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 + 1.01e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (-2.08e5 + 8.65e4i)T + (1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (-1.53e5 - 1.53e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (-1.41e3 + 584. i)T + (3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 - 9.28e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (3.47e5 - 8.38e5i)T + (-2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-2.46e6 + 2.46e6i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (-1.56e6 + 3.77e6i)T + (-8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 - 3.82e5T + 2.38e13T^{2} \) |
| 53 | \( 1 + (1.28e6 + 5.31e5i)T + (4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (1.02e7 + 4.22e6i)T + (1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (1.22e7 - 5.06e6i)T + (1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (7.09e6 + 1.71e7i)T + (-2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (-2.69e7 + 2.69e7i)T - 6.45e14iT^{2} \) |
| 73 | \( 1 + (-1.27e7 + 1.27e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 4.06e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-2.66e7 + 1.10e7i)T + (1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (1.43e7 + 1.43e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + 1.66e8T + 7.83e15T^{2} \) |
| 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
−15.49633791948690994507936888923, −13.85182133557754488381716696949, −12.22357341960911778238563659958, −10.84354923360327978276202130959, −9.493118275618244157852926992959, −9.032447517350485047221248103636, −7.34385735862118025844907950512, −5.03755436530234211616789180656, −3.14692770826721539041254278634, −1.01034738097111789466933923055,
1.33006895000913209301701700749, 2.65282457227438408376861410296, 6.07219345587934862542252812574, 7.24869451494355607578524397170, 8.245024680533068551634535226203, 9.812745471908846679495209007709, 11.07779059157357616299210406589, 12.52277584593306394835719071088, 14.05046733598731735108138752401, 14.95091164401343442354620763851
