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 | \( 1 + 5.72e4iT - 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.89888038477210399928869432852, −12.66140188837679041778709059057, −11.54780854176698316749929377090, −10.45294075503986962766315703495, −9.392247596903900042299163347082, −8.327828444785171481498386493140, −5.55076696098224587340927679165, −4.79486190249958671949713218274, −3.28848030807508636876165059055, −1.18964987684626317977382824297,
2.03107198785693670520310772779, 3.97902935339007798831911231978, 6.00838729734193831055676555834, 6.88159177392055134592565838449, 7.945502438704377770485314387004, 9.268517258977201841067485835350, 11.46920399306802280263884968833, 12.08374675598941706969939146896, 13.72001599119866577979637523377, 14.34095462690690441154529275982