| L(s) = 1 | + (−0.819 − 0.573i)2-s + (−0.117 − 1.72i)3-s + (0.342 + 0.939i)4-s + (1.38 + 1.75i)5-s + (−0.895 + 1.48i)6-s + (3.58 − 0.960i)7-s + (0.258 − 0.965i)8-s + (−2.97 + 0.404i)9-s + (−0.125 − 2.23i)10-s + (−0.985 + 0.568i)11-s + (1.58 − 0.700i)12-s + (−0.355 + 0.0310i)13-s + (−3.48 − 1.26i)14-s + (2.87 − 2.59i)15-s + (−0.766 + 0.642i)16-s + (6.65 + 4.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 − 0.405i)2-s + (−0.0675 − 0.997i)3-s + (0.171 + 0.469i)4-s + (0.618 + 0.785i)5-s + (−0.365 + 0.605i)6-s + (1.35 − 0.363i)7-s + (0.0915 − 0.341i)8-s + (−0.990 + 0.134i)9-s + (−0.0396 − 0.705i)10-s + (−0.297 + 0.171i)11-s + (0.457 − 0.202i)12-s + (−0.0985 + 0.00862i)13-s + (−0.932 − 0.339i)14-s + (0.742 − 0.670i)15-s + (−0.191 + 0.160i)16-s + (1.61 + 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24908 - 0.567236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24908 - 0.567236i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.819 + 0.573i)T \) |
| 3 | \( 1 + (0.117 + 1.72i)T \) |
| 5 | \( 1 + (-1.38 - 1.75i)T \) |
| 19 | \( 1 + (-4.31 - 0.648i)T \) |
| good | 7 | \( 1 + (-3.58 + 0.960i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.985 - 0.568i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.355 - 0.0310i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-6.65 - 4.66i)T + (5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (1.89 + 0.882i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.296 - 1.68i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.86 + 6.70i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.54 + 1.54i)T + 37iT^{2} \) |
| 41 | \( 1 + (5.26 + 6.27i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.10 + 2.84i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (3.94 + 5.62i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (0.545 + 0.254i)T + (34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (0.869 + 4.93i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (4.37 - 1.59i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (12.4 - 8.69i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-1.54 + 4.25i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.138 - 1.57i)T + (-71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (-5.07 - 6.04i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.42 + 1.99i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (9.90 + 8.31i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.12 + 10.1i)T + (-33.1 - 91.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.56636822385855606625206037032, −10.01958036515206425825976862585, −8.706207449919416778479732907380, −7.70546549020994936638795800513, −7.48259825284034889394573698905, −6.16099828630506892572979449095, −5.27302589566333948581273799700, −3.52170383797275729554844765358, −2.21290519069487435554544038491, −1.31561024294082404626491997747,
1.26817333406542637158571101283, 2.94128609993205082964101202382, 4.79657762881472657424813262580, 5.15349952845932485651271526680, 6.00900700198563256353610510567, 7.69070532428104839825407971506, 8.271470100828160759133506844250, 9.217165481527704482805233779072, 9.783592076820853369335129958139, 10.60474874947871610606578384462
