| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.397 + 1.68i)3-s + (−0.959 − 0.281i)4-s + (2.02 − 0.949i)5-s + (−1.61 − 0.633i)6-s + (−4.08 − 2.62i)7-s + (0.415 − 0.909i)8-s + (−2.68 − 1.33i)9-s + (0.651 + 2.13i)10-s + (0.475 + 3.30i)11-s + (0.856 − 1.50i)12-s + (−0.660 − 1.02i)13-s + (3.17 − 3.66i)14-s + (0.796 + 3.79i)15-s + (0.841 + 0.540i)16-s + (−1.75 − 5.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.229 + 0.973i)3-s + (−0.479 − 0.140i)4-s + (0.905 − 0.424i)5-s + (−0.658 − 0.258i)6-s + (−1.54 − 0.991i)7-s + (0.146 − 0.321i)8-s + (−0.894 − 0.446i)9-s + (0.206 + 0.676i)10-s + (0.143 + 0.996i)11-s + (0.247 − 0.434i)12-s + (−0.183 − 0.285i)13-s + (0.849 − 0.980i)14-s + (0.205 + 0.978i)15-s + (0.210 + 0.135i)16-s + (−0.424 − 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.567695 - 0.269276i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.567695 - 0.269276i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.397 - 1.68i)T \) |
| 5 | \( 1 + (-2.02 + 0.949i)T \) |
| 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
−10.03792637885312035455740526359, −9.478362545651187985792715740788, −9.089073967651144374679815044677, −7.53774522085541973306109169775, −6.67115633574720538924306828844, −5.93569272769273618828669358825, −4.86190618812593573476226360068, −4.12426391943577979780921902898, −2.77056985018142941935369465327, −0.33712968901743576905729241919,
1.69318240327094258799445609369, 2.68572998867120167647623879112, 3.58833756429140106230410771677, 5.70122461892748829003144119554, 5.97968666311132278308849659103, 6.82668978128160077324342246439, 8.179718553733029127135716433838, 9.041654433767147580542116148783, 9.703932778685052659718349941651, 10.67006837846341537533394602624
