| L(s) = 1 | + (6.33 + 7.94i)2-s + (4.01 − 5.03i)3-s + (−15.8 + 69.3i)4-s + (−53.4 + 25.7i)5-s + 65.4·6-s − 5.04·7-s + (−358. + 172. i)8-s + (44.8 + 196. i)9-s + (−542. − 261. i)10-s + (62.1 + 272. i)11-s + (286. + 358. i)12-s + (1.00e3 − 483. i)13-s + (−31.9 − 40.0i)14-s + (−85.0 + 372. i)15-s + (−1.58e3 − 765. i)16-s + (148. + 71.3i)17-s + ⋯ |
| L(s) = 1 | + (1.11 + 1.40i)2-s + (0.257 − 0.323i)3-s + (−0.494 + 2.16i)4-s + (−0.955 + 0.460i)5-s + 0.742·6-s − 0.0389·7-s + (−1.98 + 0.953i)8-s + (0.184 + 0.808i)9-s + (−1.71 − 0.826i)10-s + (0.154 + 0.678i)11-s + (0.573 + 0.718i)12-s + (1.64 − 0.793i)13-s + (−0.0435 − 0.0546i)14-s + (−0.0975 + 0.427i)15-s + (−1.55 − 0.747i)16-s + (0.124 + 0.0599i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.703720 + 2.47804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.703720 + 2.47804i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.11e4 - 4.74e3i)T \) |
| good | 2 | \( 1 + (-6.33 - 7.94i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (-4.01 + 5.03i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (53.4 - 25.7i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + 5.04T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-62.1 - 272. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-1.00e3 + 483. i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (-148. - 71.3i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (-355. + 1.55e3i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (408. + 1.79e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (929. + 1.16e3i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-5.25e3 - 6.59e3i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 + 543.T + 6.93e7T^{2} \) |
| 41 | \( 1 + (9.97e3 + 1.25e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (3.29e3 - 1.44e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-6.16e3 - 2.97e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (3.17e4 + 1.52e4i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-2.83e4 + 3.55e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-1.05e4 + 4.61e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-6.50e3 + 2.84e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-3.79e4 + 1.82e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 + 4.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (3.78e4 - 4.74e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-6.62e4 + 8.31e4i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (-4.00e3 - 1.75e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−15.67522493550071755641292961651, −14.32055733173819778646614008938, −13.40852421893708019495797296647, −12.43833142554237278036814292615, −10.92914995846682558093367365222, −8.412959879106267807703707163496, −7.56548633192620894606264586745, −6.47662363069827151554665415253, −4.80043278075485009878660953692, −3.37505609548632407212958054114,
1.11556920968023031570678378964, 3.50789642667886065935343318571, 4.14776549738854802144018210031, 6.01443148987950806821630540944, 8.548784883362676442974618359462, 9.896582371565884870553432875284, 11.38940654628706503941179563496, 11.88950448005499804034429756208, 13.17312398564714847000480706970, 14.14058562590024300333268871742
