| L(s) = 1 | + (5.17 + 8.96i)2-s + (22.2 + 38.4i)3-s + (10.4 − 18.0i)4-s − 162.·5-s + (−229. + 398. i)6-s + (171.5 − 297. i)7-s + 1.54e3·8-s + (107. − 186. i)9-s + (−840. − 1.45e3i)10-s + (2.22e3 + 3.84e3i)11-s + 924.·12-s + (2.85e3 + 7.38e3i)13-s + 3.55e3·14-s + (−3.60e3 − 6.24e3i)15-s + (6.64e3 + 1.15e4i)16-s + (−1.40e4 + 2.43e4i)17-s + ⋯ |
| L(s) = 1 | + (0.457 + 0.792i)2-s + (0.474 + 0.822i)3-s + (0.0813 − 0.140i)4-s − 0.580·5-s + (−0.434 + 0.752i)6-s + (0.188 − 0.327i)7-s + 1.06·8-s + (0.0492 − 0.0853i)9-s + (−0.265 − 0.460i)10-s + (0.503 + 0.871i)11-s + 0.154·12-s + (0.360 + 0.932i)13-s + 0.345·14-s + (−0.275 − 0.477i)15-s + (0.405 + 0.702i)16-s + (−0.694 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.39182 + 2.76210i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39182 + 2.76210i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-171.5 + 297. i)T \) |
| 13 | \( 1 + (-2.85e3 - 7.38e3i)T \) |
| good | 2 | \( 1 + (-5.17 - 8.96i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-22.2 - 38.4i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 162.T + 7.81e4T^{2} \) |
| 11 | \( 1 + (-2.22e3 - 3.84e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.40e4 - 2.43e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (9.73e3 - 1.68e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.40e4 - 2.42e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.67e4 + 2.90e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.35e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (2.29e5 + 3.98e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-9.06e4 - 1.56e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (2.64e5 - 4.57e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + 1.05e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.42e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (7.00e5 - 1.21e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.58e5 - 7.94e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.10e5 + 1.92e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.04e6 - 1.81e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 4.71e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.77e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.65e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (4.41e6 + 7.64e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.43e6 + 2.48e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.42641817317751848551186239693, −11.99536819390545945656341513428, −10.75096355290103850419396537065, −9.774193919855245046557692476915, −8.509533002998877552528191561942, −7.22758385080277437987438310642, −6.19501993810390338411605591636, −4.42005398895156658333521239474, −3.99546030107852730127332791442, −1.64370302231313078510143696717,
0.844760077676204216722270324969, 2.33125535446273769235500898793, 3.33326172469678026534019153157, 4.81181507780293873513618549549, 6.72723194630057515004018617357, 7.85440610227716075968755369910, 8.679978800951496403423377545575, 10.53836028585380975962587512222, 11.53762185857434913406085815341, 12.23166456943488596394426744484
