| L(s) = 1 | + (0.732 + 2.73i)2-s + (−3.39 + 5.87i)3-s + (−6.92 + 4i)4-s + (−10.7 + 10.7i)5-s + (−18.5 − 4.97i)6-s + (−4.14 + 15.4i)7-s + (−16 − 15.9i)8-s + (17.4 + 30.2i)9-s + (−37.3 − 21.5i)10-s + (63.4 − 16.9i)11-s − 54.3i·12-s + (147. + 82.4i)13-s − 45.2·14-s + (−26.8 − 100. i)15-s + (31.9 − 55.4i)16-s + (260. − 150. i)17-s + ⋯ |
| L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.377 + 0.653i)3-s + (−0.433 + 0.250i)4-s + (−0.431 + 0.431i)5-s + (−0.515 − 0.138i)6-s + (−0.0845 + 0.315i)7-s + (−0.250 − 0.249i)8-s + (0.215 + 0.373i)9-s + (−0.373 − 0.215i)10-s + (0.523 − 0.140i)11-s − 0.377i·12-s + (0.872 + 0.487i)13-s − 0.230·14-s + (−0.119 − 0.444i)15-s + (0.124 − 0.216i)16-s + (0.901 − 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.451465 + 1.05287i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.451465 + 1.05287i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.732 - 2.73i)T \) |
| 13 | \( 1 + (-147. - 82.4i)T \) |
| good | 3 | \( 1 + (3.39 - 5.87i)T + (-40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (10.7 - 10.7i)T - 625iT^{2} \) |
| 7 | \( 1 + (4.14 - 15.4i)T + (-2.07e3 - 1.20e3i)T^{2} \) |
| 11 | \( 1 + (-63.4 + 16.9i)T + (1.26e4 - 7.32e3i)T^{2} \) |
| 17 | \( 1 + (-260. + 150. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (6.07 + 1.62i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (336. + 194. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (72.9 - 126. i)T + (-3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (627. - 627. i)T - 9.23e5iT^{2} \) |
| 37 | \( 1 + (-1.94e3 + 521. i)T + (1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + (767. + 2.86e3i)T + (-2.44e6 + 1.41e6i)T^{2} \) |
| 43 | \( 1 + (-2.06e3 + 1.19e3i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (951. + 951. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 - 2.64e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (1.13e3 - 4.24e3i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-2.53e3 - 4.39e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.48e3 + 5.54e3i)T + (-1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 + (7.43e3 + 1.99e3i)T + (2.20e7 + 1.27e7i)T^{2} \) |
| 73 | \( 1 + (23.0 + 23.0i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 7.91e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-4.63e3 + 4.63e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (-2.69e3 + 722. i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (1.30e4 + 3.49e3i)T + (7.66e7 + 4.42e7i)T^{2} \) |
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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
−16.67962834255026847841874568216, −16.02755548661808921430052881566, −14.90574433716322290593248290238, −13.69079975816855211950569215450, −11.97959250567587927358573209159, −10.65786023368742921794268589115, −9.069110482230866161307262847485, −7.36424087778269593269153803975, −5.68512184951209486716510877138, −3.93686424710163313401917983692,
1.03903734375382019758360713863, 3.92975737993702926841939945975, 6.10574467893529052224289125262, 7.988959756878490637736437666111, 9.746990718942337220212084551544, 11.35153291481158579507121644962, 12.38116926655679257896492256256, 13.28900675348821741152248565132, 14.83087193638987749757367137472, 16.38260127580214523526903499090
