L(s) = 1 | + (3.28 + 4.60i)2-s + (3.33 − 15.2i)3-s + (−10.3 + 30.2i)4-s + (17.3 + 53.1i)5-s + (81.0 − 34.7i)6-s + 121.·7-s + (−173. + 51.9i)8-s + (−220. − 101. i)9-s + (−187. + 254. i)10-s + 442.·11-s + (426. + 258. i)12-s + 1.18e3i·13-s + (398. + 557. i)14-s + (867. − 87.4i)15-s + (−809. − 627. i)16-s + 1.03e3·17-s + ⋯ |
L(s) = 1 | + (0.581 + 0.813i)2-s + (0.213 − 0.976i)3-s + (−0.323 + 0.946i)4-s + (0.310 + 0.950i)5-s + (0.919 − 0.394i)6-s + 0.934·7-s + (−0.958 + 0.286i)8-s + (−0.908 − 0.417i)9-s + (−0.592 + 0.805i)10-s + 1.10·11-s + (0.855 + 0.518i)12-s + 1.94i·13-s + (0.543 + 0.760i)14-s + (0.994 − 0.100i)15-s + (−0.790 − 0.612i)16-s + 0.866·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.05864 + 1.63373i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.05864 + 1.63373i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.28 - 4.60i)T \) |
| 3 | \( 1 + (-3.33 + 15.2i)T \) |
| 5 | \( 1 + (-17.3 - 53.1i)T \) |
good | 7 | \( 1 - 121.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 442.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.18e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.29e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 2.19e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.78e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.53e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 4.28e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 7.10e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 194.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.08e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.52e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.29e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.41e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.27e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.33e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 3.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.10e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 5.72e4iT - 8.58e9T^{2} \) |
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
−14.34095462690690441154529275982, −13.72001599119866577979637523377, −12.08374675598941706969939146896, −11.46920399306802280263884968833, −9.268517258977201841067485835350, −7.945502438704377770485314387004, −6.88159177392055134592565838449, −6.00838729734193831055676555834, −3.97902935339007798831911231978, −2.03107198785693670520310772779,
1.18964987684626317977382824297, 3.28848030807508636876165059055, 4.79486190249958671949713218274, 5.55076696098224587340927679165, 8.327828444785171481498386493140, 9.392247596903900042299163347082, 10.45294075503986962766315703495, 11.54780854176698316749929377090, 12.66140188837679041778709059057, 13.89888038477210399928869432852