| L(s) = 1 | + (0.212 − 0.977i)2-s + (−0.800 + 0.599i)3-s + (−0.909 − 0.415i)4-s + (−2.06 + 0.850i)5-s + (0.415 + 0.909i)6-s + (−0.184 + 2.58i)7-s + (−0.599 + 0.800i)8-s + (0.281 − 0.959i)9-s + (0.391 + 2.20i)10-s + (2.58 − 4.02i)11-s + (0.977 − 0.212i)12-s + (−4.73 + 0.338i)13-s + (2.48 + 0.729i)14-s + (1.14 − 1.92i)15-s + (0.654 + 0.755i)16-s + (1.98 − 5.30i)17-s + ⋯ |
| L(s) = 1 | + (0.150 − 0.690i)2-s + (−0.462 + 0.345i)3-s + (−0.454 − 0.207i)4-s + (−0.924 + 0.380i)5-s + (0.169 + 0.371i)6-s + (−0.0697 + 0.975i)7-s + (−0.211 + 0.283i)8-s + (0.0939 − 0.319i)9-s + (0.123 + 0.696i)10-s + (0.780 − 1.21i)11-s + (0.282 − 0.0613i)12-s + (−1.31 + 0.0938i)13-s + (0.663 + 0.194i)14-s + (0.295 − 0.495i)15-s + (0.163 + 0.188i)16-s + (0.480 − 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.513924 - 0.647073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.513924 - 0.647073i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.212 + 0.977i)T \) |
| 3 | \( 1 + (0.800 - 0.599i)T \) |
| 5 | \( 1 + (2.06 - 0.850i)T \) |
| 23 | \( 1 + (-3.66 - 3.09i)T \) |
| good | 7 | \( 1 + (0.184 - 2.58i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (-2.58 + 4.02i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (4.73 - 0.338i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-1.98 + 5.30i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 3.58i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-6.23 + 2.84i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.969 + 6.74i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.64 + 3.01i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (9.76 - 2.86i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (3.97 + 5.30i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (0.388 - 0.388i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.03 - 0.502i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-0.565 - 0.490i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (7.52 - 1.08i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (11.6 + 2.52i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-6.13 + 3.94i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-9.22 + 3.43i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (3.27 - 3.77i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.43 + 1.33i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-1.69 + 11.7i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-14.8 - 8.09i)T + (52.4 + 81.6i)T^{2} \) |
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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
−10.35486135365059702354403166528, −9.387723521174372580029544440623, −8.823875766828770745872001745646, −7.60541529766271636003583396162, −6.61418298991509217793924446112, −5.43413703380443564145096351063, −4.69627142809053048292862298442, −3.42988121048225218408581697413, −2.67416165272374539619159282502, −0.50055084371206210391437926871,
1.30248898484685417109845985548, 3.49848831275128703246883213028, 4.50547975678247709222130170077, 5.11227769088494797419784860048, 6.64497289247564960045981861620, 7.09959695809295027327175455847, 7.85807869897869144504733355688, 8.723526612217997973136348270917, 10.00309691232645183955267032178, 10.51907523248125937486067091383
