L(s) = 1 | + (2.20 − 4.48i)5-s + (0.607 + 0.350i)7-s + (−0.400 − 0.231i)11-s + (12.2 − 7.05i)13-s + 14.3·17-s + 5.37·19-s + (20.8 + 36.0i)23-s + (−15.2 − 19.8i)25-s + (−8.09 − 4.67i)29-s + (−10.9 − 18.9i)31-s + (2.91 − 1.95i)35-s + 34.0i·37-s + (19.3 − 11.1i)41-s + (48.4 + 27.9i)43-s + (3.82 − 6.62i)47-s + ⋯ |
L(s) = 1 | + (0.441 − 0.897i)5-s + (0.0868 + 0.0501i)7-s + (−0.0364 − 0.0210i)11-s + (0.939 − 0.542i)13-s + 0.846·17-s + 0.283·19-s + (0.905 + 1.56i)23-s + (−0.610 − 0.792i)25-s + (−0.279 − 0.161i)29-s + (−0.352 − 0.611i)31-s + (0.0833 − 0.0558i)35-s + 0.921i·37-s + (0.472 − 0.272i)41-s + (1.12 + 0.650i)43-s + (0.0813 − 0.140i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.435976797\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.435976797\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.20 + 4.48i)T \) |
good | 7 | \( 1 + (-0.607 - 0.350i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (0.400 + 0.231i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-12.2 + 7.05i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 14.3T + 289T^{2} \) |
| 19 | \( 1 - 5.37T + 361T^{2} \) |
| 23 | \( 1 + (-20.8 - 36.0i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (8.09 + 4.67i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (10.9 + 18.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 34.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-19.3 + 11.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-48.4 - 27.9i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-3.82 + 6.62i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 36.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-30.3 + 17.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-29.4 + 50.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-53.6 + 30.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 43.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 50.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-48.0 + 83.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (13.0 - 22.5i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 125. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-102. - 58.8i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.305202049530199188183063379312, −8.199249330169668104796143307066, −7.77067278131641105703718650353, −6.57843842028788701022750108244, −5.59071468246430754253110545859, −5.23390021247693055456502381287, −4.00987399090125586040315815343, −3.10922374982306929627889879051, −1.69559537924747426174186776309, −0.803989537113729963674491198182,
1.08306739546217988567412753767, 2.32867195435301548481217099669, 3.26297464751541467729419011608, 4.19074618646514335502326382458, 5.36822341850455009812383253306, 6.13891538111575456996118770924, 6.89561532899476167048076074775, 7.59395365738509134753514826429, 8.633594056458157942364580622422, 9.303172656537354389734481462554