| L(s) = 1 | + (0.284 − 1.97i)2-s + (1.24 + 2.72i)3-s + (−3.83 − 1.12i)4-s + (−2.69 − 3.11i)5-s + (5.75 − 1.69i)6-s + (22.2 + 14.2i)7-s + (−3.32 + 7.27i)8-s + (−5.89 + 6.80i)9-s + (−6.93 + 4.45i)10-s + (3.56 + 24.7i)11-s + (−1.70 − 11.8i)12-s + (43.8 − 28.1i)13-s + (34.6 − 39.9i)14-s + (5.13 − 11.2i)15-s + (13.4 + 8.65i)16-s + (12.3 − 3.61i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.241 − 0.278i)5-s + (0.391 − 0.115i)6-s + (1.20 + 0.771i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.219 + 0.140i)10-s + (0.0976 + 0.678i)11-s + (−0.0410 − 0.285i)12-s + (0.935 − 0.601i)13-s + (0.660 − 0.762i)14-s + (0.0883 − 0.193i)15-s + (0.210 + 0.135i)16-s + (0.175 − 0.0516i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.96766 - 0.0580420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96766 - 0.0580420i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.284 + 1.97i)T \) |
| 3 | \( 1 + (-1.24 - 2.72i)T \) |
| 23 | \( 1 + (-63.9 - 89.8i)T \) |
| good | 5 | \( 1 + (2.69 + 3.11i)T + (-17.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-22.2 - 14.2i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (-3.56 - 24.7i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (-43.8 + 28.1i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-12.3 + 3.61i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (-108. - 31.7i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (34.9 - 10.2i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (54.8 - 120. i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (-91.2 + 105. i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (114. + 132. i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-52.4 - 114. i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 + 304.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (510. + 328. i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (279. - 179. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (345. - 757. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (15.7 - 109. i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-135. + 938. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (-147. - 43.4i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (290. - 186. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-552. + 638. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-493. - 1.08e3i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (743. + 857. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−12.47559059746050156435204213828, −11.64105615484231978036992110191, −10.81212455300130373786994922751, −9.624761789961882738993371802533, −8.657059407699681772210669450394, −7.74588221574867833750358241775, −5.58044556085256037013816515837, −4.67579348224530912093322095471, −3.25133924199130206952590990710, −1.54647196966367646149713177172,
1.17773416791926724482493629607, 3.48400738227245163489008774404, 4.89977590472887965490341174370, 6.34840779752754240351443845576, 7.44334250914438123678858262924, 8.174165249931429995022264524869, 9.275310742005733367095547480277, 10.97103156479416381728628836427, 11.55518294639121646096608425920, 13.08904530844622559867495038052
