| L(s) = 1 | + (−2.14 − 2.69i)2-s + (12.6 − 15.8i)3-s + (4.48 − 19.6i)4-s + (43.8 − 21.1i)5-s − 69.5·6-s + 137.·7-s + (−161. + 77.8i)8-s + (−36.9 − 161. i)9-s + (−151. − 72.7i)10-s + (52.0 + 227. i)11-s + (−254. − 318. i)12-s + (4.63 − 2.23i)13-s + (−296. − 371. i)14-s + (219. − 960. i)15-s + (−24.8 − 11.9i)16-s + (−1.21e3 − 586. i)17-s + ⋯ |
| L(s) = 1 | + (−0.379 − 0.475i)2-s + (0.808 − 1.01i)3-s + (0.140 − 0.614i)4-s + (0.785 − 0.378i)5-s − 0.788·6-s + 1.06·7-s + (−0.893 + 0.430i)8-s + (−0.151 − 0.665i)9-s + (−0.477 − 0.230i)10-s + (0.129 + 0.567i)11-s + (−0.509 − 0.638i)12-s + (0.00761 − 0.00366i)13-s + (−0.403 − 0.506i)14-s + (0.251 − 1.10i)15-s + (−0.0242 − 0.0116i)16-s + (−1.02 − 0.491i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.03014 - 1.77562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03014 - 1.77562i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-4.36e3 + 1.13e4i)T \) |
| good | 2 | \( 1 + (2.14 + 2.69i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (-12.6 + 15.8i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (-43.8 + 21.1i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 - 137.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-52.0 - 227. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-4.63 + 2.23i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (1.21e3 + 586. i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (93.4 - 409. i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (-469. - 2.05e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (-2.73e3 - 3.43e3i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-579. - 726. i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 - 3.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (9.69e3 + 1.21e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (619. - 2.71e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-2.47e4 - 1.19e4i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (2.55e4 + 1.22e4i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-2.69e4 + 3.37e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-1.14e4 + 5.00e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (1.76e4 - 7.74e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-1.99e3 + 959. i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 - 1.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (2.01e4 - 2.52e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (6.67e4 - 8.37e4i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (-4.01e4 - 1.75e5i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.26786566463328489628587812402, −13.55841037570563138947194748977, −12.20388967299380801280303266600, −10.90620802942562607729307937967, −9.470153279504678365926216202368, −8.464189230673390418865409541595, −6.96492810319156221636874218177, −5.22756355022405536965791026680, −2.22366885472263717949700017339, −1.39022799164310995625981283528,
2.64118026952540609233162454640, 4.32633686266161270298182794034, 6.39980786922422584688738101437, 8.169143842250816107365604427952, 8.928846134891977490491494796880, 10.21858793144051925415364436840, 11.52857332985959123512337813199, 13.34179985614906589085229896867, 14.51285330660352047747651918064, 15.27521502089365965823673428042
