| L(s) = 1 | + (−5.14 − 6.45i)2-s + (12.1 + 1.83i)3-s + (−8.05 + 35.2i)4-s + (6.56 + 4.47i)5-s + (−50.9 − 88.2i)6-s + (−125. + 216. i)7-s + (31.2 − 15.0i)8-s + (−86.8 − 26.7i)9-s + (−4.90 − 65.4i)10-s + (5.72 + 25.0i)11-s + (−163. + 415. i)12-s + (−50.2 + 670. i)13-s + (2.04e3 − 308. i)14-s + (71.8 + 66.6i)15-s + (785. + 378. i)16-s + (809. − 551. i)17-s + ⋯ |
| L(s) = 1 | + (−0.910 − 1.14i)2-s + (0.782 + 0.117i)3-s + (−0.251 + 1.10i)4-s + (0.117 + 0.0801i)5-s + (−0.577 − 1.00i)6-s + (−0.965 + 1.67i)7-s + (0.172 − 0.0832i)8-s + (−0.357 − 0.110i)9-s + (−0.0155 − 0.207i)10-s + (0.0142 + 0.0625i)11-s + (−0.327 + 0.833i)12-s + (−0.0824 + 1.10i)13-s + (2.78 − 0.420i)14-s + (0.0824 + 0.0765i)15-s + (0.766 + 0.369i)16-s + (0.679 − 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.591419 + 0.345351i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.591419 + 0.345351i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.09e4 + 5.21e3i)T \) |
| good | 2 | \( 1 + (5.14 + 6.45i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (-12.1 - 1.83i)T + (232. + 71.6i)T^{2} \) |
| 5 | \( 1 + (-6.56 - 4.47i)T + (1.14e3 + 2.90e3i)T^{2} \) |
| 7 | \( 1 + (125. - 216. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-5.72 - 25.0i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (50.2 - 670. i)T + (-3.67e5 - 5.53e4i)T^{2} \) |
| 17 | \( 1 + (-809. + 551. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (-46.5 + 14.3i)T + (2.04e6 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (2.66e3 - 2.46e3i)T + (4.80e5 - 6.41e6i)T^{2} \) |
| 29 | \( 1 + (-3.24e3 + 489. i)T + (1.95e7 - 6.04e6i)T^{2} \) |
| 31 | \( 1 + (1.62e3 - 4.14e3i)T + (-2.09e7 - 1.94e7i)T^{2} \) |
| 37 | \( 1 + (-2.67e3 - 4.63e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-8.10e3 - 1.01e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-1.54e3 + 6.74e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-2.23e3 - 2.97e4i)T + (-4.13e8 + 6.23e7i)T^{2} \) |
| 59 | \( 1 + (1.20e4 + 5.80e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (1.89e3 + 4.81e3i)T + (-6.19e8 + 5.74e8i)T^{2} \) |
| 67 | \( 1 + (9.98e3 - 3.08e3i)T + (1.11e9 - 7.60e8i)T^{2} \) |
| 71 | \( 1 + (-1.07e4 - 9.99e3i)T + (1.34e8 + 1.79e9i)T^{2} \) |
| 73 | \( 1 + (671. - 8.96e3i)T + (-2.04e9 - 3.08e8i)T^{2} \) |
| 79 | \( 1 + (1.00e3 - 1.74e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-9.53e4 - 1.43e4i)T + (3.76e9 + 1.16e9i)T^{2} \) |
| 89 | \( 1 + (-9.83e3 - 1.48e3i)T + (5.33e9 + 1.64e9i)T^{2} \) |
| 97 | \( 1 + (-2.21e4 - 9.68e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−15.14456893791537623576178304393, −13.91163298517994573505022174711, −12.24270767883049509893921603239, −11.70523897082219147794213217538, −9.840452498414803493559274945327, −9.265814820799088369915014303814, −8.322324934749286735386236330822, −6.02029061990967164052579456783, −3.20626825866319070743761294356, −2.18452065472446542913643932385,
0.43534763966756083618573295657, 3.45449749908369892805081607748, 6.02673145989612793956027990516, 7.41213285268711054250614672313, 8.162692910998537722227304512016, 9.576588694463068444801852467736, 10.47804284162016290425266731388, 12.80765282442706844981302372804, 13.90703896760999651685296457059, 14.87302562719408412827312173729
