L(s) = 1 | + (2.67 + 1.54i)2-s + (−2.05 + 4.77i)3-s + (0.766 + 1.32i)4-s + (−5.33 + 9.82i)5-s + (−12.8 + 9.58i)6-s + (27.1 + 15.6i)7-s − 19.9i·8-s + (−18.5 − 19.6i)9-s + (−29.4 + 18.0i)10-s + (9.59 − 16.6i)11-s + (−7.90 + 0.926i)12-s + (18.1 − 10.4i)13-s + (48.3 + 83.7i)14-s + (−35.8 − 45.6i)15-s + (36.9 − 64.0i)16-s − 6.19i·17-s + ⋯ |
L(s) = 1 | + (0.945 + 0.545i)2-s + (−0.395 + 0.918i)3-s + (0.0957 + 0.165i)4-s + (−0.477 + 0.878i)5-s + (−0.875 + 0.652i)6-s + (1.46 + 0.845i)7-s − 0.882i·8-s + (−0.686 − 0.726i)9-s + (−0.930 + 0.569i)10-s + (0.263 − 0.455i)11-s + (−0.190 + 0.0222i)12-s + (0.387 − 0.223i)13-s + (0.922 + 1.59i)14-s + (−0.617 − 0.786i)15-s + (0.577 − 1.00i)16-s − 0.0883i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0844 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0844 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.23731 + 1.34668i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23731 + 1.34668i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.05 - 4.77i)T \) |
| 5 | \( 1 + (5.33 - 9.82i)T \) |
good | 2 | \( 1 + (-2.67 - 1.54i)T + (4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-27.1 - 15.6i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-9.59 + 16.6i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-18.1 + 10.4i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 6.19iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 96.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + (141. - 81.4i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-3.82 + 6.62i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (112. + 194. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 155. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (-157. - 272. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (166. + 96.3i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (275. + 159. i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 277. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (214. + 371. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-44.9 + 77.8i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-505. + 291. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 259. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-62.3 + 107. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-31.8 - 18.3i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 333.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-891. - 514. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.35280462924104359486924864707, −14.70358239848744370183636290225, −13.88553411453803665750215669027, −11.85461170464447329383429413072, −11.22523152122503623735877764228, −9.720494243590382091663908388917, −7.989188844657848788249519332849, −6.11715564891907823786692336523, −5.08071159951933090088172969277, −3.64321678188752278531120835067,
1.53694389926733547115633303496, 4.21023314952851798511464748163, 5.31748013731029716756731138865, 7.50853076000502039303786380031, 8.451247295219951512525349143844, 10.95409243761771165273458755912, 11.81970527666769033608252712751, 12.56113478198009920851124573841, 13.77310334824096285451357243032, 14.38846660887161535695695279621