| L(s) = 1 | + (−3.16 + 5.49i)5-s + (−9.59 − 16.6i)7-s + (−20.6 − 35.7i)11-s + (−19.6 + 34.0i)13-s + 59.9·17-s − 41.2·19-s + (26.9 − 46.5i)23-s + (42.4 + 73.4i)25-s + (32.6 + 56.5i)29-s + (−32.3 + 56.0i)31-s + 121.·35-s + 293.·37-s + (−103. + 179. i)41-s + (188. + 327. i)43-s + (−161. − 280. i)47-s + ⋯ |
| L(s) = 1 | + (−0.283 + 0.491i)5-s + (−0.518 − 0.897i)7-s + (−0.565 − 0.979i)11-s + (−0.418 + 0.725i)13-s + 0.855·17-s − 0.498·19-s + (0.243 − 0.422i)23-s + (0.339 + 0.587i)25-s + (0.208 + 0.361i)29-s + (−0.187 + 0.324i)31-s + 0.587·35-s + 1.30·37-s + (−0.394 + 0.683i)41-s + (0.670 + 1.16i)43-s + (−0.502 − 0.870i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.157195866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157195866\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.16 - 5.49i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (9.59 + 16.6i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (20.6 + 35.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (19.6 - 34.0i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 59.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 41.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-26.9 + 46.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-32.6 - 56.5i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (32.3 - 56.0i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (103. - 179. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-188. - 327. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (161. + 280. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 340.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (203. - 352. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-378. - 655. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (422. - 731. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 859.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 789.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-108. - 188. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (50.5 + 87.6i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-873. - 1.51e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−10.41286782006288906710526065901, −9.586909582902069167486862556856, −8.525812368041821980910122301297, −7.58171083167133847546817023168, −6.85769723928780487016815153554, −5.94835901826922570844542070685, −4.70425143764376854575069550699, −3.61198345803263148162091057227, −2.75859456120374215506547610568, −0.977240130025775541286392381900,
0.40177654498541656422163315448, 2.12481382406000742816044590671, 3.16664043136713148527575556155, 4.54021624454533421971080879685, 5.37581581246757951493926685819, 6.29758485725068696306667270823, 7.53445520526706946524400354471, 8.157388862132681195346077043039, 9.234418400210187543818699639461, 9.860987679225818799584220812110
