| L(s) = 1 | + (0.483 + 1.32i)2-s + (−1.53 + 1.28i)4-s + (7.71 − 1.35i)5-s + (−0.690 − 0.579i)7-s + (−2.44 − 1.41i)8-s + (5.53 + 9.59i)10-s + (15.2 + 2.68i)11-s + (−0.854 − 0.310i)13-s + (0.435 − 1.19i)14-s + (0.694 − 3.93i)16-s + (−10.6 + 6.15i)17-s + (−5.40 + 9.36i)19-s + (−10.0 + 11.9i)20-s + (3.80 + 21.5i)22-s + (21.0 + 25.1i)23-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (−0.383 + 0.321i)4-s + (1.54 − 0.271i)5-s + (−0.0986 − 0.0827i)7-s + (−0.306 − 0.176i)8-s + (0.553 + 0.959i)10-s + (1.38 + 0.244i)11-s + (−0.0657 − 0.0239i)13-s + (0.0311 − 0.0855i)14-s + (0.0434 − 0.246i)16-s + (−0.627 + 0.362i)17-s + (−0.284 + 0.492i)19-s + (−0.503 + 0.599i)20-s + (0.172 + 0.979i)22-s + (0.917 + 1.09i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.82322 + 0.855534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82322 + 0.855534i\) |
| \(L(2)\) |
|
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 | \( 1 + (-7.71 + 1.35i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (0.690 + 0.579i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-15.2 - 2.68i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (0.854 + 0.310i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (10.6 - 6.15i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (5.40 - 9.36i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-21.0 - 25.1i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (19.3 + 53.1i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (37.9 - 31.8i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (17.4 + 30.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-12.2 + 33.6i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-7.20 + 40.8i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-15.9 + 18.9i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 50.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (65.7 - 11.5i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (18.7 + 15.7i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (61.2 + 22.3i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (24.4 - 14.1i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (10.7 - 18.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (27.2 - 9.90i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-0.0509 - 0.139i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-79.3 - 45.7i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (17.4 - 98.9i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.06973938364816821787060803027, −12.03259616231448575679808758356, −10.60196151879653054873901790946, −9.376217932864011313979933842237, −8.957997199605547057652847051390, −7.25938752285206373027231549590, −6.23721652136726827101619755909, −5.40752965737242410966917942827, −3.91840860398039051927423591036, −1.81370171821695499283523299050,
1.58667715165617167554623871697, 2.97144390461470189691783090540, 4.67221611170801767535692179412, 5.99319895361685630509657294594, 6.84409446781040615467720190286, 9.018820194075245349370318968049, 9.345909024355404759764737338073, 10.60469299801443300092962252790, 11.34414038992908738062577169137, 12.68514522087641107378573429662
