| L(s) = 1 | + (0.445 − 1.94i)2-s + (−2.69 + 3.38i)3-s + (−3.60 − 1.73i)4-s + (4.53 − 5.69i)5-s + (5.39 + 6.77i)6-s + (18.2 + 3.26i)7-s + (−4.98 + 6.25i)8-s + (1.83 + 8.04i)9-s + (−9.07 − 11.3i)10-s + (14.6 − 64.2i)11-s + (15.6 − 7.51i)12-s + (9.14 − 40.0i)13-s + (14.4 − 34.0i)14-s + (7.01 + 30.7i)15-s + (9.97 + 12.5i)16-s + (68.4 − 32.9i)17-s + ⋯ |
| L(s) = 1 | + (0.157 − 0.689i)2-s + (−0.519 + 0.651i)3-s + (−0.450 − 0.216i)4-s + (0.405 − 0.508i)5-s + (0.367 + 0.460i)6-s + (0.984 + 0.176i)7-s + (−0.220 + 0.276i)8-s + (0.0679 + 0.297i)9-s + (−0.287 − 0.359i)10-s + (0.401 − 1.76i)11-s + (0.375 − 0.180i)12-s + (0.195 − 0.855i)13-s + (0.276 − 0.650i)14-s + (0.120 + 0.528i)15-s + (0.155 + 0.195i)16-s + (0.976 − 0.470i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.42028 - 0.772102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42028 - 0.772102i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.445 + 1.94i)T \) |
| 7 | \( 1 + (-18.2 - 3.26i)T \) |
| good | 3 | \( 1 + (2.69 - 3.38i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-4.53 + 5.69i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-14.6 + 64.2i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-9.14 + 40.0i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-68.4 + 32.9i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 - 60.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-28.7 - 13.8i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (121. - 58.5i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 - 71.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + (268. - 129. i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (213. - 267. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-160. - 200. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-125. + 551. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-386. - 185. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (463. + 581. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 4.96i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + 780.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-485. - 233. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (58.8 + 257. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + 391.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (0.899 + 3.93i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-299. - 1.31e3i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 677.T + 9.12e5T^{2} \) |
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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
−13.34574988495337923557999534851, −11.87888799276552676446292545019, −11.17212902200475567399473245442, −10.30074062023933034937581932405, −9.065477044056224520854177123119, −7.990715794515615103326361731273, −5.60051158731063021776606462167, −5.11255579227140385309346639377, −3.37709749415850221088008155982, −1.15925006205604051885627217571,
1.64396299322481743754417333507, 4.22726237117390559680253368534, 5.65700397114155799442333124502, 6.90163431154251944412841472001, 7.50805818900615597990502419226, 9.153888161792565479859667251628, 10.34417763996254796186085408315, 11.83907869560031844692399415665, 12.42652105461341077559274508839, 13.85180965752111218667791843437
