| L(s) = 1 | + (0.997 − 0.0713i)2-s + (0.212 − 0.977i)3-s + (0.989 − 0.142i)4-s + (0.855 + 2.06i)5-s + (0.142 − 0.989i)6-s + (1.43 − 2.61i)7-s + (0.977 − 0.212i)8-s + (−0.909 − 0.415i)9-s + (1.00 + 1.99i)10-s + (1.89 − 1.64i)11-s + (0.0713 − 0.997i)12-s + (−0.493 + 0.269i)13-s + (1.23 − 2.71i)14-s + (2.20 − 0.396i)15-s + (0.959 − 0.281i)16-s + (−3.79 − 5.07i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0504i)2-s + (0.122 − 0.564i)3-s + (0.494 − 0.0711i)4-s + (0.382 + 0.923i)5-s + (0.0580 − 0.404i)6-s + (0.540 − 0.990i)7-s + (0.345 − 0.0751i)8-s + (−0.303 − 0.138i)9-s + (0.316 + 0.632i)10-s + (0.570 − 0.494i)11-s + (0.0205 − 0.287i)12-s + (−0.136 + 0.0747i)13-s + (0.331 − 0.725i)14-s + (0.568 − 0.102i)15-s + (0.239 − 0.0704i)16-s + (−0.920 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.57567 - 0.807646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.57567 - 0.807646i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 + 0.0713i)T \) |
| 3 | \( 1 + (-0.212 + 0.977i)T \) |
| 5 | \( 1 + (-0.855 - 2.06i)T \) |
| 23 | \( 1 + (-4.79 + 0.201i)T \) |
| good | 7 | \( 1 + (-1.43 + 2.61i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.89 + 1.64i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.493 - 0.269i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (3.79 + 5.07i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-1.00 - 6.97i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-8.91 - 1.28i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (3.85 + 2.47i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.50 - 4.03i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (2.06 + 4.51i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.505 - 0.109i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (3.07 + 3.07i)T + 47iT^{2} \) |
| 53 | \( 1 + (12.4 + 6.79i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (1.37 - 4.67i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.62 + 2.52i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.11 - 15.5i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (6.74 - 7.77i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (8.43 + 6.31i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (0.503 + 0.147i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.74 + 1.02i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (8.73 - 5.61i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (5.02 + 1.87i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.59470311455013103975487545675, −9.735829039652695100504172658750, −8.486973672211916866321655501972, −7.39920160321178876714944970135, −6.87426994738300434010806361793, −6.05227161710819213964258488402, −4.87128194972684558027299755798, −3.71706661111657991355643811658, −2.72585335208820104525877977288, −1.38315768874775892605185667108,
1.73575377580004083582076145677, 2.90773736001586956661664571795, 4.51384788445265794488678972632, 4.79943254528092560307287216160, 5.85324733555256365978535307148, 6.77136550848997006698570880523, 8.178075360113445696806800803295, 8.917252260608019013184429874344, 9.474870292020000055206287297595, 10.73345520182447853036243672903
