| L(s) = 1 | + (−1.53 − 1.28i)2-s + (4.44 + 2.68i)3-s + (0.694 + 3.93i)4-s + (3.40 + 1.23i)5-s + (−3.36 − 9.83i)6-s + (−3.51 + 19.9i)7-s + (4.00 − 6.92i)8-s + (12.6 + 23.8i)9-s + (−3.62 − 6.27i)10-s + (49.0 − 17.8i)11-s + (−7.47 + 19.3i)12-s + (20.2 − 16.9i)13-s + (31.0 − 26.0i)14-s + (11.8 + 14.6i)15-s + (−15.0 + 5.47i)16-s + (−23.7 − 41.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.856 + 0.516i)3-s + (0.0868 + 0.492i)4-s + (0.304 + 0.110i)5-s + (−0.229 − 0.668i)6-s + (−0.189 + 1.07i)7-s + (0.176 − 0.306i)8-s + (0.466 + 0.884i)9-s + (−0.114 − 0.198i)10-s + (1.34 − 0.488i)11-s + (−0.179 + 0.466i)12-s + (0.432 − 0.362i)13-s + (0.592 − 0.496i)14-s + (0.203 + 0.252i)15-s + (−0.234 + 0.0855i)16-s + (−0.338 − 0.586i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.42223 + 0.294663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42223 + 0.294663i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.53 + 1.28i)T \) |
| 3 | \( 1 + (-4.44 - 2.68i)T \) |
| good | 5 | \( 1 + (-3.40 - 1.23i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (3.51 - 19.9i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-49.0 + 17.8i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-20.2 + 16.9i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (23.7 + 41.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (61.3 - 106. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (17.4 + 98.9i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (66.6 + 55.9i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (39.5 + 224. i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-60.3 - 104. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (24.5 - 20.5i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-267. + 97.2i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-43.5 + 247. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 747.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-573. - 208. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-25.6 + 145. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-680. + 571. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (186. + 323. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (315. - 547. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-812. - 682. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (931. + 781. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (150. - 261. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-424. + 154. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−14.92189380021802921091109778179, −13.90850169303791814823907793426, −12.56884780001825720910125495375, −11.32881596512064951679635235911, −9.956272351808186036320378392411, −9.054711090518335023116484686893, −8.160685489334177496136621225465, −6.14436759372818303112595710895, −3.88298820929161536245091232617, −2.25440409436477218565920829459,
1.48905244464250599364676311813, 4.00124964665626054022521241943, 6.50540426440031005353670125701, 7.33146714996331482796451313177, 8.829282501667729938910436409089, 9.629676696171492722906055125416, 11.14342353821219023067856482879, 12.84167815534774142387317788387, 13.83822361634726315425301471553, 14.66739752220856246127201459269
