| L(s) = 1 | + (−1.37 − 0.499i)2-s + (2.68 − 1.34i)3-s + (−1.43 − 1.20i)4-s + (1.75 + 4.68i)5-s + (−4.35 + 0.499i)6-s + (5.58 + 6.65i)7-s + (4.28 + 7.42i)8-s + (5.40 − 7.19i)9-s + (−0.0649 − 7.30i)10-s + (9.52 − 1.67i)11-s + (−5.44 − 1.30i)12-s + (−3.36 − 9.23i)13-s + (−4.34 − 11.9i)14-s + (10.9 + 10.2i)15-s + (−0.875 − 4.96i)16-s + (1.70 − 2.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.686 − 0.249i)2-s + (0.894 − 0.446i)3-s + (−0.357 − 0.300i)4-s + (0.350 + 0.936i)5-s + (−0.725 + 0.0832i)6-s + (0.798 + 0.951i)7-s + (0.535 + 0.927i)8-s + (0.600 − 0.799i)9-s + (−0.00649 − 0.730i)10-s + (0.865 − 0.152i)11-s + (−0.454 − 0.108i)12-s + (−0.258 − 0.710i)13-s + (−0.310 − 0.851i)14-s + (0.732 + 0.681i)15-s + (−0.0547 − 0.310i)16-s + (0.100 − 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.38495 - 0.200931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38495 - 0.200931i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.68 + 1.34i)T \) |
| 5 | \( 1 + (-1.75 - 4.68i)T \) |
| good | 2 | \( 1 + (1.37 + 0.499i)T + (3.06 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-5.58 - 6.65i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-9.52 + 1.67i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (3.36 + 9.23i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 2.94i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (12.3 + 21.4i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-29.3 - 24.6i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (1.94 - 5.33i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-24.4 - 20.4i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (38.2 + 22.0i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-14.4 - 39.7i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (55.2 - 9.73i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (11.3 - 9.55i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 47.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (70.9 + 12.5i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-73.4 + 61.6i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (2.22 + 6.12i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (109. + 63.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-21.4 + 12.3i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-94.3 - 34.3i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (61.4 + 22.3i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-46.1 + 26.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (60.0 - 10.5i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.14526021801973781723656674222, −11.71321851700953714127254913969, −10.79240616476607346610740982997, −9.544060294494101259945843026741, −8.893334105377662348534556513731, −7.898140884168960405723415454076, −6.62348882157437443047070128832, −5.08087240405465306279176066037, −2.98306637237756241767840025210, −1.64051574785373684230175097576,
1.46901978062035621752229112575, 4.01240441223658800737350673696, 4.67196678961794219933042675053, 6.94719335384850659716279412961, 8.137528661234689111172580478938, 8.729400727119117029752922809615, 9.660986260835215004213160189147, 10.52341792796264370938873549914, 12.17289510208976689192954719471, 13.26904168415998337390802298300
