| L(s) = 1 | + (0.215 − 0.590i)3-s + (−1.30 − 7.40i)5-s + (3.03 − 5.25i)7-s + (6.59 + 5.53i)9-s + (4.46 + 7.74i)11-s + (−6.31 − 17.3i)13-s + (−4.65 − 0.821i)15-s + (−15.3 + 12.9i)17-s + (17.8 + 6.56i)19-s + (−2.45 − 2.92i)21-s + (−4.31 + 24.4i)23-s + (−29.6 + 10.7i)25-s + (9.58 − 5.53i)27-s + (−13.6 + 16.2i)29-s + (34.7 + 20.0i)31-s + ⋯ |
| L(s) = 1 | + (0.0716 − 0.196i)3-s + (−0.261 − 1.48i)5-s + (0.433 − 0.750i)7-s + (0.732 + 0.614i)9-s + (0.406 + 0.703i)11-s + (−0.485 − 1.33i)13-s + (−0.310 − 0.0547i)15-s + (−0.904 + 0.759i)17-s + (0.938 + 0.345i)19-s + (−0.116 − 0.139i)21-s + (−0.187 + 1.06i)23-s + (−1.18 + 0.431i)25-s + (0.355 − 0.204i)27-s + (−0.469 + 0.559i)29-s + (1.12 + 0.646i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.11375 - 0.646260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11375 - 0.646260i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-17.8 - 6.56i)T \) |
| good | 3 | \( 1 + (-0.215 + 0.590i)T + (-6.89 - 5.78i)T^{2} \) |
| 5 | \( 1 + (1.30 + 7.40i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-3.03 + 5.25i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.46 - 7.74i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (6.31 + 17.3i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (15.3 - 12.9i)T + (50.1 - 284. i)T^{2} \) |
| 23 | \( 1 + (4.31 - 24.4i)T + (-497. - 180. i)T^{2} \) |
| 29 | \( 1 + (13.6 - 16.2i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-34.7 - 20.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 14.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-19.0 + 52.2i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-6.72 - 38.1i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-1.10 - 0.927i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + (-23.5 - 4.14i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (34.9 + 41.7i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (0.501 - 2.84i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-71.6 + 85.3i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (114. - 20.1i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (42.9 + 15.6i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (5.39 - 14.8i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (38.7 - 67.0i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-49.3 - 135. i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (110. + 131. i)T + (-1.63e3 + 9.26e3i)T^{2} \) |
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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
−13.84190366462563869399603853429, −12.90208951662513868068165635847, −12.22460032086713907901116265523, −10.70306199396540686555814410756, −9.548799892599367909922820018612, −8.159197214197374447819567084080, −7.35964302675419197610339944782, −5.24730230837861649891221984938, −4.21580593301898221239494291031, −1.35592296945481226204145089743,
2.66074595420451147009624700959, 4.31894637530831421158241605278, 6.34722017135137815677255008553, 7.22161331052549974396349687549, 8.896236531331665597477253437059, 9.967552980420195438513292879774, 11.38960115793048828676409613761, 11.82823503277192830316221079007, 13.65154617451744679573275221514, 14.58200637726308821815429764136
