| L(s) = 1 | + (−2.53 − 0.921i)2-s + (2.82 + 1.01i)3-s + (2.49 + 2.09i)4-s + (4.53 − 2.11i)5-s + (−6.21 − 5.17i)6-s + (−6.84 − 8.16i)7-s + (0.993 + 1.72i)8-s + (6.93 + 5.74i)9-s + (−13.4 + 1.16i)10-s + (8.07 − 1.42i)11-s + (4.92 + 8.46i)12-s + (1.66 + 4.58i)13-s + (9.81 + 26.9i)14-s + (14.9 − 1.34i)15-s + (−3.19 − 18.1i)16-s + (9.31 − 16.1i)17-s + ⋯ |
| L(s) = 1 | + (−1.26 − 0.460i)2-s + (0.940 + 0.339i)3-s + (0.624 + 0.524i)4-s + (0.906 − 0.422i)5-s + (−1.03 − 0.862i)6-s + (−0.978 − 1.16i)7-s + (0.124 + 0.215i)8-s + (0.770 + 0.637i)9-s + (−1.34 + 0.116i)10-s + (0.733 − 0.129i)11-s + (0.410 + 0.705i)12-s + (0.128 + 0.352i)13-s + (0.701 + 1.92i)14-s + (0.995 − 0.0897i)15-s + (−0.199 − 1.13i)16-s + (0.547 − 0.948i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.946889 - 0.559449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.946889 - 0.559449i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.82 - 1.01i)T \) |
| 5 | \( 1 + (-4.53 + 2.11i)T \) |
| good | 2 | \( 1 + (2.53 + 0.921i)T + (3.06 + 2.57i)T^{2} \) |
| 7 | \( 1 + (6.84 + 8.16i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-8.07 + 1.42i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-1.66 - 4.58i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-9.31 + 16.1i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (8.83 + 15.3i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-11.0 - 9.24i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-8.37 + 23.0i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (10.8 + 9.10i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-51.1 - 29.5i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-9.94 - 27.3i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (32.6 - 5.75i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (49.9 - 41.9i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 63.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-50.2 - 8.85i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (87.7 - 73.6i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-17.4 - 47.9i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (0.0479 + 0.0276i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (8.05 - 4.64i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (102. + 37.2i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-136. - 49.6i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (33.1 - 19.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-140. + 24.7i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.12173266607433244245488443469, −11.38853111980497790060509120888, −10.21607105538359641194016883750, −9.573099951373462396955422116126, −9.110004110046064445529146160556, −7.82540789780737868424741880018, −6.64556511714841827268518547787, −4.55803395510917438507311839117, −2.86926914433994361127979624008, −1.15945074035692240977658754393,
1.78660342488167172678504360623, 3.37005642923972086783970533313, 6.07317306412772392657642912254, 6.77329219803456254869042510308, 8.142848791172870959966279962270, 9.036731332845404712790044197503, 9.616125138959873635553199400768, 10.50956544059410743948607505429, 12.49483618221521954405901002731, 13.02763936936613099721516036066
