L(s) = 1 | + (−1.11 − 0.875i)2-s + (−0.900 − 0.433i)3-s + (0.465 + 1.94i)4-s + (0.698 − 1.45i)5-s + (0.620 + 1.27i)6-s + (1.90 − 1.84i)7-s + (1.18 − 2.56i)8-s + (0.623 + 0.781i)9-s + (−2.04 + 0.998i)10-s + (0.316 + 0.252i)11-s + (0.424 − 1.95i)12-s + (4.51 + 3.60i)13-s + (−3.72 + 0.378i)14-s + (−1.25 + 1.00i)15-s + (−3.56 + 1.81i)16-s + (0.218 + 0.0499i)17-s + ⋯ |
L(s) = 1 | + (−0.785 − 0.619i)2-s + (−0.520 − 0.250i)3-s + (0.232 + 0.972i)4-s + (0.312 − 0.648i)5-s + (0.253 + 0.518i)6-s + (0.718 − 0.695i)7-s + (0.419 − 0.907i)8-s + (0.207 + 0.260i)9-s + (−0.647 + 0.315i)10-s + (0.0955 + 0.0762i)11-s + (0.122 − 0.564i)12-s + (1.25 + 0.999i)13-s + (−0.994 + 0.101i)14-s + (−0.325 + 0.259i)15-s + (−0.891 + 0.452i)16-s + (0.0530 + 0.0121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0745 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0745 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.746082 - 0.692406i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.746082 - 0.692406i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.11 + 0.875i)T \) |
| 3 | \( 1 + (0.900 + 0.433i)T \) |
| 7 | \( 1 + (-1.90 + 1.84i)T \) |
good | 5 | \( 1 + (-0.698 + 1.45i)T + (-3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-0.316 - 0.252i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-4.51 - 3.60i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.218 - 0.0499i)T + (15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 0.409T + 19T^{2} \) |
| 23 | \( 1 + (-0.361 + 0.0825i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-1.16 + 5.08i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 + (-1.78 + 7.82i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (-2.15 + 4.48i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-3.03 - 6.30i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-1.19 + 1.49i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (2.01 + 8.82i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (4.95 - 2.38i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (6.33 + 1.44i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 + (-3.45 + 0.787i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (9.75 - 7.78i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 4.62iT - 79T^{2} \) |
| 83 | \( 1 + (0.804 + 1.00i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-2.38 + 1.90i)T + (19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 - 11.4iT - 97T^{2} \) |
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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
−10.71043329492738839380185522263, −9.571397011978701819481237227341, −8.859081636225121188679435609091, −7.963698436941797325603662759934, −7.09412864384222224027266146150, −6.05512640530091582987328818265, −4.68045048224977303762983589081, −3.79777949034173931419665322118, −1.92659992601181852837334097476, −0.977699225954070745370150224614,
1.32019734738277648506749992950, 2.92997625741896107035958535367, 4.71503705587369225251116064861, 5.75546033331544783617702396925, 6.24220379794092300909887203736, 7.37046842461425170124152774933, 8.352620566540314756156847269452, 9.013652930505479560865066585716, 10.13470601270202843083032699746, 10.77912189207482120341106326391