| L(s) = 1 | + (−0.959 − 0.281i)2-s + (0.0953 − 0.663i)3-s + (0.841 + 0.540i)4-s + (−0.415 − 0.909i)5-s + (−0.278 + 0.609i)6-s + (4.29 + 1.26i)7-s + (−0.654 − 0.755i)8-s + (2.44 + 0.718i)9-s + (0.142 + 0.989i)10-s + (1.96 + 4.30i)11-s + (0.438 − 0.506i)12-s + (−1.28 + 1.48i)13-s + (−3.76 − 2.41i)14-s + (−0.642 + 0.188i)15-s + (0.415 + 0.909i)16-s + (−5.78 + 3.71i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (0.0550 − 0.382i)3-s + (0.420 + 0.270i)4-s + (−0.185 − 0.406i)5-s + (−0.113 + 0.248i)6-s + (1.62 + 0.476i)7-s + (−0.231 − 0.267i)8-s + (0.815 + 0.239i)9-s + (0.0450 + 0.313i)10-s + (0.592 + 1.29i)11-s + (0.126 − 0.146i)12-s + (−0.356 + 0.411i)13-s + (−1.00 − 0.646i)14-s + (−0.166 + 0.0487i)15-s + (0.103 + 0.227i)16-s + (−1.40 + 0.901i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29360 + 0.175933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29360 + 0.175933i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 67 | \( 1 + (-7.22 + 3.85i)T \) |
| good | 3 | \( 1 + (-0.0953 + 0.663i)T + (-2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (-4.29 - 1.26i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.96 - 4.30i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (1.28 - 1.48i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.78 - 3.71i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (2.99 - 0.878i)T + (15.9 - 10.2i)T^{2} \) |
| 23 | \( 1 + (0.256 - 1.78i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 - 0.923T + 29T^{2} \) |
| 31 | \( 1 + (-3.84 - 4.44i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 + (-3.39 + 2.17i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-4.52 + 2.90i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (0.545 - 3.79i)T + (-45.0 - 13.2i)T^{2} \) |
| 53 | \( 1 + (11.2 + 7.21i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (0.502 + 0.580i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.48 - 3.25i)T + (-39.9 - 46.1i)T^{2} \) |
| 71 | \( 1 + (9.04 + 5.81i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.53 + 14.3i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.29 + 3.79i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (5.44 + 11.9i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-1.98 - 13.8i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + 6.82T + 97T^{2} \) |
| 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
−10.61581489224249171268342815935, −9.562168367348005158257530393152, −8.742445743328542139873869767942, −8.021863114189131280710513818305, −7.27477458775622526513538454952, −6.36174566742221807749735667154, −4.72102371279142960591325255448, −4.32180309036430729548137738641, −2.03179854199783202695888653182, −1.65280810271215931229640560048,
0.960649401876098543348930188943, 2.49871989376283233961428266878, 4.08995079767462169579676192867, 4.82192082020981839645998351198, 6.21819666634017397035187808846, 7.09964672233790298710365494348, 7.965724244201719593191370094723, 8.674165521819846536851007272249, 9.558519017536489384859859061218, 10.59146299648560887916308328091
