| L(s) = 1 | + (0.764 + 1.18i)2-s + (−0.830 + 1.81i)4-s + (−0.404 − 1.37i)5-s + (−1.64 − 1.89i)7-s + (−2.79 + 0.402i)8-s + (1.32 − 1.53i)10-s + (−5.25 + 8.17i)11-s + (−7.25 + 8.36i)13-s + (0.999 − 3.40i)14-s + (−2.61 − 3.02i)16-s + (−13.6 + 6.24i)17-s + (−5.06 + 11.0i)19-s + (2.84 + 0.408i)20-s − 13.7·22-s + (−10.9 + 20.2i)23-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (−0.207 + 0.454i)4-s + (−0.0808 − 0.275i)5-s + (−0.234 − 0.270i)7-s + (−0.349 + 0.0503i)8-s + (0.132 − 0.153i)10-s + (−0.477 + 0.742i)11-s + (−0.557 + 0.643i)13-s + (0.0713 − 0.243i)14-s + (−0.163 − 0.188i)16-s + (−0.804 + 0.367i)17-s + (−0.266 + 0.583i)19-s + (0.142 + 0.0204i)20-s − 0.624·22-s + (−0.476 + 0.879i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0645i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0262776 + 0.813882i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0262776 + 0.813882i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.764 - 1.18i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (10.9 - 20.2i)T \) |
| good | 5 | \( 1 + (0.404 + 1.37i)T + (-21.0 + 13.5i)T^{2} \) |
| 7 | \( 1 + (1.64 + 1.89i)T + (-6.97 + 48.5i)T^{2} \) |
| 11 | \( 1 + (5.25 - 8.17i)T + (-50.2 - 110. i)T^{2} \) |
| 13 | \( 1 + (7.25 - 8.36i)T + (-24.0 - 167. i)T^{2} \) |
| 17 | \( 1 + (13.6 - 6.24i)T + (189. - 218. i)T^{2} \) |
| 19 | \( 1 + (5.06 - 11.0i)T + (-236. - 272. i)T^{2} \) |
| 29 | \( 1 + (47.2 - 21.5i)T + (550. - 635. i)T^{2} \) |
| 31 | \( 1 + (1.03 + 7.19i)T + (-922. + 270. i)T^{2} \) |
| 37 | \( 1 + (3.01 + 0.884i)T + (1.15e3 + 740. i)T^{2} \) |
| 41 | \( 1 + (-2.24 - 7.63i)T + (-1.41e3 + 908. i)T^{2} \) |
| 43 | \( 1 + (-5.87 + 40.8i)T + (-1.77e3 - 520. i)T^{2} \) |
| 47 | \( 1 - 2.35iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (23.7 - 20.6i)T + (399. - 2.78e3i)T^{2} \) |
| 59 | \( 1 + (-10.3 - 8.92i)T + (495. + 3.44e3i)T^{2} \) |
| 61 | \( 1 + (-7.14 - 49.6i)T + (-3.57e3 + 1.04e3i)T^{2} \) |
| 67 | \( 1 + (12.0 - 7.72i)T + (1.86e3 - 4.08e3i)T^{2} \) |
| 71 | \( 1 + (-35.7 - 55.6i)T + (-2.09e3 + 4.58e3i)T^{2} \) |
| 73 | \( 1 + (4.40 - 9.64i)T + (-3.48e3 - 4.02e3i)T^{2} \) |
| 79 | \( 1 + (-69.3 + 80.0i)T + (-888. - 6.17e3i)T^{2} \) |
| 83 | \( 1 + (14.9 - 50.8i)T + (-5.79e3 - 3.72e3i)T^{2} \) |
| 89 | \( 1 + (43.0 + 6.19i)T + (7.60e3 + 2.23e3i)T^{2} \) |
| 97 | \( 1 + (87.4 - 25.6i)T + (7.91e3 - 5.08e3i)T^{2} \) |
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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
−11.60410987248392645340937689174, −10.48945977814953758460420315186, −9.528710770652740514456926007024, −8.632823233567470762534790422396, −7.53551670610356273084492231625, −6.87041036132707516405681023294, −5.69963082494294979607514251846, −4.66984994118900456474125757784, −3.74574201981789870889364968316, −2.07665771626944010760062182649,
0.28667023103688883918588336190, 2.33373294820088246977917214061, 3.25161096303693849081658831722, 4.59646608306506866071391511674, 5.61446912550567545380082863382, 6.63418057339987997350097213784, 7.82674909615793244515709420923, 8.909079011208135388575909585758, 9.771530040635869875457656724439, 10.85873453965705933454855850656
