L(s) = 1 | + (2.66 − 2.66i)2-s + (−2.12 + 2.12i)3-s − 10.1i·4-s + (4.54 − 2.09i)5-s + 11.3i·6-s + (−2.95 + 2.95i)7-s + (−16.4 − 16.4i)8-s − 0.0323i·9-s + (6.51 − 17.6i)10-s + 8.43i·11-s + (21.6 + 21.6i)12-s + (8.44 + 9.88i)13-s + 15.7i·14-s + (−5.19 + 14.0i)15-s − 47.1·16-s + (−9.34 − 9.34i)17-s + ⋯ |
L(s) = 1 | + (1.33 − 1.33i)2-s + (−0.708 + 0.708i)3-s − 2.54i·4-s + (0.908 − 0.418i)5-s + 1.88i·6-s + (−0.421 + 0.421i)7-s + (−2.06 − 2.06i)8-s − 0.00359i·9-s + (0.651 − 1.76i)10-s + 0.767i·11-s + (1.80 + 1.80i)12-s + (0.649 + 0.760i)13-s + 1.12i·14-s + (−0.346 + 0.939i)15-s − 2.94·16-s + (−0.549 − 0.549i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.46092 - 1.29312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46092 - 1.29312i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-4.54 + 2.09i)T \) |
| 13 | \( 1 + (-8.44 - 9.88i)T \) |
good | 2 | \( 1 + (-2.66 + 2.66i)T - 4iT^{2} \) |
| 3 | \( 1 + (2.12 - 2.12i)T - 9iT^{2} \) |
| 7 | \( 1 + (2.95 - 2.95i)T - 49iT^{2} \) |
| 11 | \( 1 - 8.43iT - 121T^{2} \) |
| 17 | \( 1 + (9.34 + 9.34i)T + 289iT^{2} \) |
| 19 | \( 1 - 12.7T + 361T^{2} \) |
| 23 | \( 1 + (28.0 - 28.0i)T - 529iT^{2} \) |
| 29 | \( 1 + 32.7iT - 841T^{2} \) |
| 31 | \( 1 + 36.2iT - 961T^{2} \) |
| 37 | \( 1 + (17.1 - 17.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 26.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (34.4 - 34.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-9.09 + 9.09i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-31.0 + 31.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 68.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 30.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-5.29 + 5.29i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 2.71iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-29.3 - 29.3i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 12.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-3.27 - 3.27i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 68.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + (51.9 - 51.9i)T - 9.40e3iT^{2} \) |
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
−13.83839210457462770519929728393, −13.32769273903643372872162582892, −12.01198872181141544536719492361, −11.33576464528644076165842506498, −9.988912659173787763791681594103, −9.525325785024477592733798093970, −6.16015731260573374857793049133, −5.25972332391076866638648092838, −4.14101494656099380112780966424, −2.09889802689806441833167946224,
3.44184367845712095410047564393, 5.46527758807266511503545883105, 6.28762198467612897024397217567, 6.99150657089397946799711797513, 8.541280956581085102824600548933, 10.66626303759709224180048954891, 12.16508971551873080298520273828, 13.07080211496424673324724350195, 13.74389096734844193286921819168, 14.68346105574521943552428099476