| L(s) = 1 | + (−1.53 + 1.28i)2-s + (0.694 − 3.93i)4-s + (−1.07 + 0.391i)5-s + (0.470 + 2.66i)7-s + (4.00 + 6.92i)8-s + (1.14 − 1.98i)10-s + (−11.5 − 4.21i)11-s + (32.4 + 27.2i)13-s + (−4.14 − 3.48i)14-s + (−15.0 − 5.47i)16-s + (−21.7 + 37.6i)17-s + (65.6 + 113. i)19-s + (0.795 + 4.51i)20-s + (23.1 − 8.43i)22-s + (4.11 − 23.3i)23-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.0868 − 0.492i)4-s + (−0.0963 + 0.0350i)5-s + (0.0253 + 0.144i)7-s + (0.176 + 0.306i)8-s + (0.0362 − 0.0627i)10-s + (−0.317 − 0.115i)11-s + (0.693 + 0.581i)13-s + (−0.0792 − 0.0664i)14-s + (−0.234 − 0.0855i)16-s + (−0.310 + 0.537i)17-s + (0.792 + 1.37i)19-s + (0.00889 + 0.0504i)20-s + (0.224 − 0.0817i)22-s + (0.0373 − 0.211i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.538532 + 0.811428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.538532 + 0.811428i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + (1.07 - 0.391i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-0.470 - 2.66i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (11.5 + 4.21i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-32.4 - 27.2i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (21.7 - 37.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-65.6 - 113. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-4.11 + 23.3i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (134. - 112. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (33.0 - 187. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (26.4 - 45.7i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-179. - 150. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (365. + 132. i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-89.7 - 509. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 97.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-677. + 246. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (86.3 + 489. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-106. - 89.0i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-488. + 846. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-209. - 363. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (203. - 170. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-466. + 391. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (532. + 923. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (829. + 302. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−12.71758473217570516482397488134, −11.54351367051043899892665235579, −10.62392620339457326413486650606, −9.550530073927070945128230477195, −8.547571151505294821029096627266, −7.60409692767952498935135091589, −6.39570896825079255964391364148, −5.32497297076369315741329103586, −3.64630014767647493748478006656, −1.60393032262742170898828454418,
0.56341069877352163564144092016, 2.46722332094416857040849637661, 3.94658312448767031615400616213, 5.49629740934234035649147391786, 7.03996649686423062720441042191, 8.016549007501776458577012713561, 9.122618077831500902329833747053, 10.04999769909160772072204389534, 11.13899212845664853081477999979, 11.80157663089188600497926035425
