L(s) = 1 | + (−1.05 − 0.946i)2-s + (0.207 + 1.98i)4-s + 1.60i·5-s + (1.66 − 2.28i)8-s + (1.52 − 1.68i)10-s + 2.67i·11-s − 3.37i·13-s + (−3.91 + 0.823i)16-s + 1.60i·17-s + 4.30·19-s + (−3.19 + 0.333i)20-s + (2.53 − 2.81i)22-s − 6.46i·23-s + 2.41·25-s + (−3.19 + 3.54i)26-s + ⋯ |
L(s) = 1 | + (−0.742 − 0.669i)2-s + (0.103 + 0.994i)4-s + 0.719i·5-s + (0.588 − 0.808i)8-s + (0.481 − 0.534i)10-s + 0.807i·11-s − 0.937i·13-s + (−0.978 + 0.205i)16-s + 0.390i·17-s + 0.987·19-s + (−0.715 + 0.0744i)20-s + (0.540 − 0.599i)22-s − 1.34i·23-s + 0.482·25-s + (−0.627 + 0.696i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.152642045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.152642045\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.05 + 0.946i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.60iT - 5T^{2} \) |
| 11 | \( 1 - 2.67iT - 11T^{2} \) |
| 13 | \( 1 + 3.37iT - 13T^{2} \) |
| 17 | \( 1 - 1.60iT - 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 + 6.46iT - 23T^{2} \) |
| 29 | \( 1 + 4.71T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 0.242T + 37T^{2} \) |
| 41 | \( 1 - 11.6iT - 41T^{2} \) |
| 43 | \( 1 + 7.95iT - 43T^{2} \) |
| 47 | \( 1 + 9.04T + 47T^{2} \) |
| 53 | \( 1 - 2.46T + 53T^{2} \) |
| 59 | \( 1 - 9.04T + 59T^{2} \) |
| 61 | \( 1 - 3.37iT - 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 - 8.03iT - 71T^{2} \) |
| 73 | \( 1 - 6.17iT - 73T^{2} \) |
| 79 | \( 1 - 3.29iT - 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 0.275iT - 89T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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
−9.577000751991018397131329630352, −8.484585913750471812072314827282, −7.946596433251158777914436092613, −7.05903690381299823316098340272, −6.45660603040764437245353380046, −5.13615739731791336809376321045, −4.11190054236956735136051756090, −3.06332760131381896592381244837, −2.40825013383874555099721681010, −1.00208866235109440875137000474,
0.72300504249796505748268645515, 1.80127367765379861982832855577, 3.29947295213422328202575762615, 4.61960908776387475507874317852, 5.31320317642040125709717337699, 6.12908233719349292959856797787, 6.98727532732142873574952705847, 7.78605189264587363402919530083, 8.486454372817289158239362277670, 9.288838748527821898962241800889