| L(s) = 1 | + (−2.30 − 7.08i)2-s + (0.135 + 0.0439i)3-s + (−31.9 + 23.2i)4-s − 18.1·5-s − 1.05i·6-s + (10.5 − 7.69i)7-s + (141. + 102. i)8-s + (−65.5 − 47.5i)9-s + (41.7 + 128. i)10-s + (43.9 + 60.5i)11-s + (−5.34 + 1.73i)12-s + (−266. − 86.6i)13-s + (−78.8 − 57.3i)14-s + (−2.45 − 0.798i)15-s + (207. − 637. i)16-s + (56.3 − 77.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.575 − 1.77i)2-s + (0.0150 + 0.00488i)3-s + (−1.99 + 1.45i)4-s − 0.726·5-s − 0.0294i·6-s + (0.216 − 0.157i)7-s + (2.20 + 1.60i)8-s + (−0.808 − 0.587i)9-s + (0.417 + 1.28i)10-s + (0.363 + 0.500i)11-s + (−0.0370 + 0.0120i)12-s + (−1.57 − 0.512i)13-s + (−0.402 − 0.292i)14-s + (−0.0109 − 0.00354i)15-s + (0.809 − 2.49i)16-s + (0.195 − 0.268i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 - 0.911i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.412 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.178643 + 0.276933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.178643 + 0.276933i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-518. + 809. i)T \) |
| good | 2 | \( 1 + (2.30 + 7.08i)T + (-12.9 + 9.40i)T^{2} \) |
| 3 | \( 1 + (-0.135 - 0.0439i)T + (65.5 + 47.6i)T^{2} \) |
| 5 | \( 1 + 18.1T + 625T^{2} \) |
| 7 | \( 1 + (-10.5 + 7.69i)T + (741. - 2.28e3i)T^{2} \) |
| 11 | \( 1 + (-43.9 - 60.5i)T + (-4.52e3 + 1.39e4i)T^{2} \) |
| 13 | \( 1 + (266. + 86.6i)T + (2.31e4 + 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-56.3 + 77.6i)T + (-2.58e4 - 7.94e4i)T^{2} \) |
| 19 | \( 1 + (160. + 493. i)T + (-1.05e5 + 7.66e4i)T^{2} \) |
| 23 | \( 1 + (349. - 481. i)T + (-8.64e4 - 2.66e5i)T^{2} \) |
| 29 | \( 1 + (-1.58e3 + 513. i)T + (5.72e5 - 4.15e5i)T^{2} \) |
| 37 | \( 1 - 601. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (547. + 1.68e3i)T + (-2.28e6 + 1.66e6i)T^{2} \) |
| 43 | \( 1 + (-705. + 229. i)T + (2.76e6 - 2.00e6i)T^{2} \) |
| 47 | \( 1 + (-542. + 1.67e3i)T + (-3.94e6 - 2.86e6i)T^{2} \) |
| 53 | \( 1 + (651. - 896. i)T + (-2.43e6 - 7.50e6i)T^{2} \) |
| 59 | \( 1 + (1.01e3 - 3.11e3i)T + (-9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 - 2.48e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 6.64e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (5.60e3 + 4.06e3i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (1.03e3 + 1.42e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (350. - 481. i)T + (-1.20e7 - 3.70e7i)T^{2} \) |
| 83 | \( 1 + (-8.29e3 + 2.69e3i)T + (3.83e7 - 2.78e7i)T^{2} \) |
| 89 | \( 1 + (-1.44e3 - 1.98e3i)T + (-1.93e7 + 5.96e7i)T^{2} \) |
| 97 | \( 1 + (-3.87e3 + 2.81e3i)T + (2.73e7 - 8.41e7i)T^{2} \) |
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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
−15.18911691000276761520635208342, −13.69472767938577645180812043218, −12.06617437852244875167785813733, −11.78432526720690466938381200910, −10.27053102498199946060663139145, −9.136641958696322850302909964838, −7.74398231186942266197934360347, −4.43923295088971986638141586972, −2.73833986452707429664198985434, −0.29032956454309304634584701830,
4.75257677020704580942001988655, 6.29953898553129758782550403462, 7.81485231204885887399739744469, 8.567517940237597520987097911342, 10.17653455721543946596150054813, 12.09003262754810707122724269308, 14.21717912954087347645153159627, 14.57122071104631968598058686918, 16.04243482027635496348634719381, 16.72390083045924792113345744585
