L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.724 − 1.57i)3-s + (0.999 + 1.73i)4-s + (0.224 − 2.43i)6-s + 2.82i·8-s + (−1.94 + 2.28i)9-s + (−5.72 − 3.30i)11-s + (2 − 2.82i)12-s + (−2.00 + 3.46i)16-s + 2.36i·17-s + (−4 + 1.41i)18-s + 6.34·19-s + (−4.67 − 8.09i)22-s + (4.44 − 2.04i)24-s + (2.5 − 4.33i)25-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)2-s + (−0.418 − 0.908i)3-s + (0.499 + 0.866i)4-s + (0.0917 − 0.995i)6-s + 0.999i·8-s + (−0.649 + 0.760i)9-s + (−1.72 − 0.996i)11-s + (0.577 − 0.816i)12-s + (−0.500 + 0.866i)16-s + 0.574i·17-s + (−0.942 + 0.333i)18-s + 1.45·19-s + (−0.996 − 1.72i)22-s + (0.908 − 0.418i)24-s + (0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20457 + 0.0997875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20457 + 0.0997875i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (0.724 + 1.57i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.72 + 3.30i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2.36iT - 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (-9.39 + 5.42i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.17 - 10.6i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (1.62 - 0.937i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.174 + 0.301i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.44 - 1.41i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 + (-4.84 + 8.39i)T + (-48.5 - 84.0i)T^{2} \) |
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
−14.40965945685779600935486563878, −13.46041437988458620714018389853, −12.80600452042692713469846878027, −11.66256123931092962844101076801, −10.65776290524712935405876278964, −8.323805465493111519116727905457, −7.51439919872580222149725574821, −6.10263812145192357411800228002, −5.14872745705443002593593979642, −2.87003400110696441359977158822,
3.00141070934747919454022704351, 4.73303866172930221653669861932, 5.54245330779234409972191726848, 7.33584879936360968511900110449, 9.487939876288600041548634650951, 10.32980201828229267574607016979, 11.31416558398520816406580231266, 12.38232301493602413733278662323, 13.46430198330321125217758542966, 14.70411958085262587315876972994