| L(s) = 1 | + (−0.579 + 1.78i)2-s + (−11.3 − 8.22i)3-s + (23.0 + 16.7i)4-s + (−55.7 − 4.12i)5-s + (21.2 − 15.4i)6-s − 222.·7-s + (−91.8 + 66.7i)8-s + (−14.5 − 44.7i)9-s + (39.6 − 97.0i)10-s + (29.0 − 89.5i)11-s + (−123. − 379. i)12-s + (152. + 470. i)13-s + (128. − 396. i)14-s + (597. + 505. i)15-s + (215. + 664. i)16-s + (−393. + 286. i)17-s + ⋯ |
| L(s) = 1 | + (−0.102 + 0.315i)2-s + (−0.726 − 0.527i)3-s + (0.720 + 0.523i)4-s + (−0.997 − 0.0737i)5-s + (0.240 − 0.175i)6-s − 1.71·7-s + (−0.507 + 0.368i)8-s + (−0.0598 − 0.184i)9-s + (0.125 − 0.307i)10-s + (0.0725 − 0.223i)11-s + (−0.246 − 0.759i)12-s + (0.250 + 0.772i)13-s + (0.175 − 0.540i)14-s + (0.685 + 0.579i)15-s + (0.210 + 0.648i)16-s + (−0.330 + 0.240i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.00293762 - 0.0566593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00293762 - 0.0566593i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (55.7 + 4.12i)T \) |
| good | 2 | \( 1 + (0.579 - 1.78i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (11.3 + 8.22i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 222.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-29.0 + 89.5i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-152. - 470. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (393. - 286. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-345. + 250. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-1.17e3 + 3.61e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (2.90e3 + 2.11e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (6.99e3 - 5.08e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (404. + 1.24e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.32e3 - 4.07e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.64e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.30e3 + 1.67e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-4.71e3 - 3.42e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.44e4 + 4.45e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.72e4 - 5.30e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-3.08e4 + 2.23e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (5.50e4 + 3.99e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (380. - 1.17e3i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.10e4 - 8.06e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-1.90e4 + 1.38e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (3.19e4 - 9.82e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (3.41e4 + 2.47e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.68572282056369656069500636298, −16.33418191303853194383394712195, −15.10721753192638254356956345501, −12.89533396795325575308952218998, −12.14072335406601064824728962920, −11.04149212107887553561271904241, −8.890665443068653810585702443715, −7.06277110058524266388620751257, −6.34082250637445726007243007464, −3.41894076165074261941870687711,
0.04030721064847231442193462885, 3.36005675223489690045217725622, 5.68891067405373776432811544851, 7.18852574113894160605494922281, 9.584153154610289928188433422340, 10.71917942349339367025923306955, 11.69193767412081157415470692349, 13.00777368679287885567919471357, 15.23052095311773501504857707647, 15.89721700964282059178456139826
