| L(s) = 1 | + (1.73 + i)2-s + (5.18 − 0.262i)3-s + (1.99 + 3.46i)4-s + (−2.24 + 3.88i)5-s + (9.25 + 4.73i)6-s + (−9.71 − 15.7i)7-s + 7.99i·8-s + (26.8 − 2.72i)9-s + (−7.77 + 4.49i)10-s + (−20.2 + 11.7i)11-s + (11.2 + 17.4i)12-s + 5.91i·13-s + (−1.05 − 37.0i)14-s + (−10.6 + 20.7i)15-s + (−8 + 13.8i)16-s + (−58.0 − 100. i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.998 − 0.0504i)3-s + (0.249 + 0.433i)4-s + (−0.200 + 0.347i)5-s + (0.629 + 0.322i)6-s + (−0.524 − 0.851i)7-s + 0.353i·8-s + (0.994 − 0.100i)9-s + (−0.245 + 0.142i)10-s + (−0.555 + 0.320i)11-s + (0.271 + 0.419i)12-s + 0.126i·13-s + (−0.0201 − 0.706i)14-s + (−0.183 + 0.357i)15-s + (−0.125 + 0.216i)16-s + (−0.828 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.07141 + 0.502335i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07141 + 0.502335i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.73 - i)T \) |
| 3 | \( 1 + (-5.18 + 0.262i)T \) |
| 7 | \( 1 + (9.71 + 15.7i)T \) |
| good | 5 | \( 1 + (2.24 - 3.88i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (20.2 - 11.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 5.91iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (58.0 + 100. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-8.02 - 4.63i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (107. + 62.1i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 207. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-122. + 70.8i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (149. - 259. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 508.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 391.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-40.2 + 69.7i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (258. - 149. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (102. + 177. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (543. + 313. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-51.3 - 89.0i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 46.9iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (228. - 131. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-533. + 924. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 270.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (443. - 768. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 219. iT - 9.12e5T^{2} \) |
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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
−15.53751756278437576330947001203, −14.28740001626504845936051691639, −13.57601155951363895051380082142, −12.50034453718687106665562720408, −10.76081424696427724305437292687, −9.355787035924827876942544780922, −7.70866109949286510570465048854, −6.80219174002168041977567325141, −4.45882152661551908755427190304, −2.91819872575656085997738859166,
2.45680303265975874847664943562, 4.11717668506248095010404118767, 6.04011309678976025256053819713, 8.031508510337169780630353496706, 9.225705255618122011282463084271, 10.58374079259204875246263388161, 12.29208193352127900892099821511, 13.06042186110944777274355090291, 14.18496079396892830517680888940, 15.45104547350635853217237624977
