| L(s) = 1 | + (−0.548 − 3.96i)2-s + (1.05 − 0.186i)3-s + (−15.3 + 4.34i)4-s + (1.47 − 1.23i)5-s + (−1.31 − 4.08i)6-s + (−26.6 − 15.3i)7-s + (25.6 + 58.6i)8-s + (−75.0 + 27.3i)9-s + (−5.71 − 5.16i)10-s + (−29.9 + 17.2i)11-s + (−15.4 + 7.46i)12-s + (−42.0 + 238. i)13-s + (−46.2 + 113. i)14-s + (1.32 − 1.58i)15-s + (218. − 133. i)16-s + (−280. − 102. i)17-s + ⋯ |
| L(s) = 1 | + (−0.137 − 0.990i)2-s + (0.117 − 0.0207i)3-s + (−0.962 + 0.271i)4-s + (0.0589 − 0.0495i)5-s + (−0.0366 − 0.113i)6-s + (−0.543 − 0.313i)7-s + (0.400 + 0.916i)8-s + (−0.926 + 0.337i)9-s + (−0.0571 − 0.0516i)10-s + (−0.247 + 0.142i)11-s + (−0.107 + 0.0518i)12-s + (−0.248 + 1.41i)13-s + (−0.236 + 0.580i)14-s + (0.00590 − 0.00704i)15-s + (0.852 − 0.522i)16-s + (−0.970 − 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0989 - 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0989 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.177931 + 0.196505i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.177931 + 0.196505i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.548 + 3.96i)T \) |
| 19 | \( 1 + (-242. - 267. i)T \) |
| good | 3 | \( 1 + (-1.05 + 0.186i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-1.47 + 1.23i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (26.6 + 15.3i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (29.9 - 17.2i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (42.0 - 238. i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (280. + 102. i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-421. + 502. i)T + (-4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (1.21e3 - 443. i)T + (5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (961. + 555. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 198.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (143. + 815. i)T + (-2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (2.29e3 + 2.73e3i)T + (-5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-173. - 475. i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-1.20e3 - 1.00e3i)T + (1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-438. + 1.20e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (674. + 566. i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-1.26e3 - 3.46e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-2.36e3 - 2.81e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-313. - 1.77e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (4.38e3 - 772. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-6.44e3 - 3.72e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (61.4 - 348. i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-1.92e3 - 701. i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−13.78347872516231624274990002333, −12.96472695525855900902438161318, −11.68844055807060871154026484793, −10.89948844083169229246414708527, −9.548717377797044240088913771078, −8.761655822905619074962979655738, −7.17859003482583193268312809505, −5.25838111270245597199017489754, −3.68888732269044395440177593210, −2.11658094723330139823848542262,
0.13390689204892826993814526561, 3.21173034641910880903838201826, 5.22071477555152457595804337528, 6.23522006395104929929819520045, 7.64021899415578013166089490880, 8.786821233303806494215303954339, 9.733341562872773041518715560229, 11.16553280967316812834609756649, 12.80302712085579315544872990800, 13.54310866275122229924478509015
