| L(s) = 1 | + (−0.483 + 1.32i)2-s + (−1.53 − 1.28i)4-s + (−5.90 − 1.04i)5-s + (5.59 − 4.69i)7-s + (2.44 − 1.41i)8-s + (4.23 − 7.34i)10-s + (20.3 − 3.59i)11-s + (6.40 − 2.33i)13-s + (3.53 + 9.71i)14-s + (0.694 + 3.93i)16-s + (−1.96 − 1.13i)17-s + (−12.1 − 21.0i)19-s + (7.70 + 9.18i)20-s + (−5.08 + 28.8i)22-s + (15.5 − 18.5i)23-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (−0.383 − 0.321i)4-s + (−1.18 − 0.208i)5-s + (0.799 − 0.671i)7-s + (0.306 − 0.176i)8-s + (0.423 − 0.734i)10-s + (1.85 − 0.326i)11-s + (0.492 − 0.179i)13-s + (0.252 + 0.693i)14-s + (0.0434 + 0.246i)16-s + (−0.115 − 0.0667i)17-s + (−0.640 − 1.10i)19-s + (0.385 + 0.459i)20-s + (−0.231 + 1.31i)22-s + (0.675 − 0.804i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.13436 - 0.143127i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13436 - 0.143127i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.483 - 1.32i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (5.90 + 1.04i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-5.59 + 4.69i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-20.3 + 3.59i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-6.40 + 2.33i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (1.96 + 1.13i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (12.1 + 21.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-15.5 + 18.5i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (6.32 - 17.3i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-16.6 - 13.9i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-22.3 + 38.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (17.5 + 48.1i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-6.41 - 36.3i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (8.15 + 9.71i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 17.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (64.9 + 11.4i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-32.3 + 27.1i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (111. - 40.4i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-46.1 - 26.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-33.8 - 58.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-104. - 38.0i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (8.63 - 23.7i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-35.4 + 20.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-7.02 - 39.8i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.54749117773083976331804259405, −11.41396249528649661110585025360, −10.82175427227807860215250399959, −9.102639957486657138498262294336, −8.450299242836033062184076167142, −7.34999558388685052016286457799, −6.46137042195075181998164065340, −4.68593935492406937523419957946, −3.85649869259552418905433693230, −0.917573289299448589156075720894,
1.58677183281009353831060977191, 3.57108496939080009962470212107, 4.49481897288646824311345685230, 6.32419747150586200739115629354, 7.75307881247515744592011863712, 8.592427465950654744334933834524, 9.574471644087030378881154991018, 11.01120456481533582363754231092, 11.78161296249444412521088494158, 12.07672175298656563627616030718
