| L(s) = 1 | + (−0.142 − 0.989i)2-s + (1.53 − 0.803i)3-s + (−0.959 + 0.281i)4-s + (1.22 + 1.86i)5-s + (−1.01 − 1.40i)6-s + (4.08 − 2.62i)7-s + (0.415 + 0.909i)8-s + (1.70 − 2.46i)9-s + (1.67 − 1.48i)10-s + (−0.475 + 3.30i)11-s + (−1.24 + 1.20i)12-s + (0.660 − 1.02i)13-s + (−3.17 − 3.66i)14-s + (3.38 + 1.87i)15-s + (0.841 − 0.540i)16-s + (−1.75 + 5.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.885 − 0.464i)3-s + (−0.479 + 0.140i)4-s + (0.549 + 0.835i)5-s + (−0.413 − 0.573i)6-s + (1.54 − 0.991i)7-s + (0.146 + 0.321i)8-s + (0.569 − 0.822i)9-s + (0.529 − 0.468i)10-s + (−0.143 + 0.996i)11-s + (−0.359 + 0.347i)12-s + (0.183 − 0.285i)13-s + (−0.849 − 0.980i)14-s + (0.874 + 0.485i)15-s + (0.210 − 0.135i)16-s + (−0.424 + 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.03438 - 1.09111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03438 - 1.09111i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-1.53 + 0.803i)T \) |
| 5 | \( 1 + (-1.22 - 1.86i)T \) |
| 23 | \( 1 + (4.18 - 2.33i)T \) |
| good | 7 | \( 1 + (-4.08 + 2.62i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.475 - 3.30i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.660 + 1.02i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.75 - 5.96i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.670 - 2.28i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.83 + 6.25i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.18 + 4.79i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.549 + 0.633i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (4.44 + 3.84i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.0334 - 0.0733i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 4.46T + 47T^{2} \) |
| 53 | \( 1 + (5.83 + 9.08i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.51 + 3.90i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (9.97 - 4.55i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.376 - 2.62i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (8.42 - 1.21i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (0.938 + 3.19i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (3.34 - 5.19i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (10.3 - 8.97i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (2.16 - 4.74i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-4.90 + 5.65i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.24871700690710166017695336466, −9.786592771222855684526527143006, −8.456445006538242321430525006364, −7.83057393469545711151438107374, −7.15552206086425770173945320285, −5.89525595980307906880102614263, −4.38449918995666210552250545263, −3.68140524497895172824864286559, −2.14024397206660734943649766205, −1.62449726786162834942355214932,
1.58097514308362569618870326679, 2.85607349881139143761955838452, 4.60585318146684701777492160685, 4.99329723106014909049101989800, 5.94019802789884805312446359528, 7.37324390125903610151162871429, 8.314483280911231884171353906644, 8.832204949676576940233183313931, 9.193106201804777013353233730875, 10.42698835579559394311298157006
