| L(s) = 1 | + (0.125 + 0.0910i)2-s + (−0.913 − 0.406i)3-s + (−0.610 − 1.87i)4-s + (1.58 − 2.73i)5-s + (−0.0774 − 0.134i)6-s + (−1.81 + 2.01i)7-s + (0.190 − 0.585i)8-s + (0.669 + 0.743i)9-s + (0.447 − 0.199i)10-s + (5.26 + 1.11i)11-s + (−0.206 + 1.96i)12-s + (−0.209 − 1.99i)13-s + (−0.410 + 0.0871i)14-s + (−2.55 + 1.85i)15-s + (−3.12 + 2.26i)16-s + (−1.94 + 0.413i)17-s + ⋯ |
| L(s) = 1 | + (0.0886 + 0.0643i)2-s + (−0.527 − 0.234i)3-s + (−0.305 − 0.939i)4-s + (0.707 − 1.22i)5-s + (−0.0316 − 0.0547i)6-s + (−0.684 + 0.759i)7-s + (0.0673 − 0.207i)8-s + (0.223 + 0.247i)9-s + (0.141 − 0.0630i)10-s + (1.58 + 0.337i)11-s + (−0.0596 + 0.567i)12-s + (−0.0580 − 0.552i)13-s + (−0.109 + 0.0232i)14-s + (−0.660 + 0.479i)15-s + (−0.780 + 0.566i)16-s + (−0.471 + 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.777144 - 0.469008i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.777144 - 0.469008i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.913 + 0.406i)T \) |
| 31 | \( 1 + (-5.43 + 1.22i)T \) |
| good | 2 | \( 1 + (-0.125 - 0.0910i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-1.58 + 2.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.81 - 2.01i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (-5.26 - 1.11i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.209 + 1.99i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (1.94 - 0.413i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.840 - 7.99i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.18 + 3.65i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.87 - 3.54i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (3.16 + 5.47i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.61 - 0.721i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.00951 + 0.0905i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (3.08 - 2.23i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.56 - 5.07i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (8.32 + 3.70i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + 7.13T + 61T^{2} \) |
| 67 | \( 1 + (1.81 - 3.13i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.92 + 5.47i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (3.00 + 0.639i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (5.51 - 1.17i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-4.04 + 1.80i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (1.85 + 5.69i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.732 - 2.25i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.81376067652389828281343968412, −12.63493491873988922477048646865, −12.14146911633152813089851357149, −10.40983052286839106141844475861, −9.462378848370968630865892210029, −8.678236656408396004318619436505, −6.41493747955866396380678819068, −5.77731163951970990139627948488, −4.52350039950212687091245613215, −1.46080609527359328594047134284,
3.08175883201146314990419834863, 4.38211526420399940987015018266, 6.58734074538465245541576015764, 6.90230037447865257882210282718, 8.953684254771499977919827156448, 9.922438606239116566689547539767, 11.15444714757502681217744915775, 11.90340678783675154408554933078, 13.52387402148010665352476708078, 13.80831444782563467524162097032
