| L(s) = 1 | + (−1 + 1.73i)2-s + (−1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (−10.4 + 18.0i)5-s + 6·6-s + (−18.3 + 2.59i)7-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (−20.8 − 36.0i)10-s + (−7.58 − 13.1i)11-s + (−6.00 + 10.3i)12-s + 2.16·13-s + (13.8 − 34.3i)14-s + 62.5·15-s + (−8 + 13.8i)16-s + (59.6 + 103. i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.931 + 1.61i)5-s + 0.408·6-s + (−0.990 + 0.140i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.658 − 1.14i)10-s + (−0.207 − 0.359i)11-s + (−0.144 + 0.249i)12-s + 0.0461·13-s + (0.264 − 0.655i)14-s + 1.07·15-s + (−0.125 + 0.216i)16-s + (0.851 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0708735 + 0.439905i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0708735 + 0.439905i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 + (18.3 - 2.59i)T \) |
| good | 5 | \( 1 + (10.4 - 18.0i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (7.58 + 13.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 2.16T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-59.6 - 103. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-16.7 + 29.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (0.325 - 0.564i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-111. - 193. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (84.2 - 145. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 323.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 221.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (254. - 439. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-88.2 - 152. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (227. + 393. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (19.3 - 33.4i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (70.8 + 122. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 602.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-551. - 954. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-58.1 + 100. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 568.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-191. + 331. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 334.T + 9.12e5T^{2} \) |
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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
−15.91768146279149837393198075725, −15.09341694490042941913029883420, −13.98596845067316145228808820572, −12.50229527888853378919430921389, −11.11101224817046887291525549230, −10.11224012363320507584594790054, −8.163339661400542710803417806359, −7.02002881950550454801392308947, −6.09982045050864616557324800905, −3.33872862581885107678518207610,
0.41850663403096990490243996388, 3.73622577202310628140833981999, 5.16171461966220405252643999397, 7.63038525208443911588343014653, 9.068926542399521256886882367120, 9.908676799935076287744025910980, 11.62411833615134343538942354176, 12.36125749380002901092411792638, 13.41060021625310863451117669834, 15.47340277292088298264307366055
