| 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
−13.16123182493738275457600843599, −12.25598874329179822180191451357, −10.09568588303056187442865900310, −9.312021361649158077384963770553, −8.574809564519466968779347636591, −6.88977414922169415167180891979, −6.66143660721006962142136847936, −5.61312610166515755036212404796, −4.51042307053402558332106481625, −3.09996582314977538150757484788,
0.22990811514159860369744480995, 1.96085224712197449171322293670, 3.06767549869182330016436822691, 4.05846885628510945448708169037, 5.66473361818970852746893554490, 6.30397889903034295812616194059, 8.743188889176079815126124797899, 9.595239483754819163600773633967, 10.15447634761654632059845704663, 10.97307909461547060838542478364
