| L(s) = 1 | + (1.71 − 5.26i)2-s + (−18.3 − 13.3i)4-s + (11.1 + 1.07i)5-s − 30.2·7-s + (−65.5 + 47.6i)8-s + (24.7 − 56.7i)10-s + (6.53 − 20.1i)11-s + (−1.85 − 5.71i)13-s + (−51.6 + 159. i)14-s + (82.6 + 254. i)16-s + (−17.3 + 12.6i)17-s + (−85.2 + 61.9i)19-s + (−189. − 167. i)20-s + (−94.7 − 68.8i)22-s + (−1.91 + 5.90i)23-s + ⋯ |
| L(s) = 1 | + (0.604 − 1.86i)2-s + (−2.28 − 1.66i)4-s + (0.995 + 0.0963i)5-s − 1.63·7-s + (−2.89 + 2.10i)8-s + (0.781 − 1.79i)10-s + (0.179 − 0.551i)11-s + (−0.0396 − 0.121i)13-s + (−0.986 + 3.03i)14-s + (1.29 + 3.97i)16-s + (−0.248 + 0.180i)17-s + (−1.02 + 0.748i)19-s + (−2.11 − 1.87i)20-s + (−0.917 − 0.666i)22-s + (−0.0173 + 0.0535i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.564592 + 0.435923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.564592 + 0.435923i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-11.1 - 1.07i)T \) |
| good | 2 | \( 1 + (-1.71 + 5.26i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 + 30.2T + 343T^{2} \) |
| 11 | \( 1 + (-6.53 + 20.1i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (1.85 + 5.71i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (17.3 - 12.6i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (85.2 - 61.9i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (1.91 - 5.90i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (187. + 135. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-142. + 103. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (90.6 + 278. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (18.2 + 56.0i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 379.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-131. - 95.7i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (214. + 155. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-141. - 436. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-185. + 570. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (269. - 195. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (715. + 519. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-100. + 310. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (63.7 + 46.3i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (1.06e3 - 771. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-348. + 1.07e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-40.0 - 29.0i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−10.97307909461547060838542478364, −10.15447634761654632059845704663, −9.595239483754819163600773633967, −8.743188889176079815126124797899, −6.30397889903034295812616194059, −5.66473361818970852746893554490, −4.05846885628510945448708169037, −3.06767549869182330016436822691, −1.96085224712197449171322293670, −0.22990811514159860369744480995,
3.09996582314977538150757484788, 4.51042307053402558332106481625, 5.61312610166515755036212404796, 6.66143660721006962142136847936, 6.88977414922169415167180891979, 8.574809564519466968779347636591, 9.312021361649158077384963770553, 10.09568588303056187442865900310, 12.25598874329179822180191451357, 13.16123182493738275457600843599
