| L(s) = 1 | + (0.392 − 0.0187i)2-s + (−0.984 − 1.42i)3-s + (−1.83 + 0.175i)4-s + (−1.77 − 1.69i)5-s + (−0.413 − 0.541i)6-s + (0.708 + 3.67i)7-s + (−1.49 + 0.215i)8-s + (−1.05 + 2.80i)9-s + (−0.729 − 0.632i)10-s + (−1.19 + 0.615i)11-s + (2.05 + 2.44i)12-s + (−4.80 − 0.925i)13-s + (0.347 + 1.43i)14-s + (−0.663 + 4.20i)15-s + (3.03 − 0.585i)16-s + (−1.26 − 2.76i)17-s + ⋯ |
| L(s) = 1 | + (0.277 − 0.0132i)2-s + (−0.568 − 0.822i)3-s + (−0.918 + 0.0877i)4-s + (−0.794 − 0.757i)5-s + (−0.168 − 0.220i)6-s + (0.267 + 1.38i)7-s + (−0.529 + 0.0760i)8-s + (−0.353 + 0.935i)9-s + (−0.230 − 0.200i)10-s + (−0.360 + 0.185i)11-s + (0.594 + 0.705i)12-s + (−1.33 − 0.256i)13-s + (0.0927 + 0.382i)14-s + (−0.171 + 1.08i)15-s + (0.759 − 0.146i)16-s + (−0.306 − 0.670i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0156537 + 0.0870090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0156537 + 0.0870090i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.984 + 1.42i)T \) |
| 23 | \( 1 + (-2.11 + 4.30i)T \) |
| good | 2 | \( 1 + (-0.392 + 0.0187i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (1.77 + 1.69i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.708 - 3.67i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (1.19 - 0.615i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (4.80 + 0.925i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (1.26 + 2.76i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (4.64 + 2.11i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.131 + 1.38i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (0.267 - 0.210i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (2.58 - 8.79i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (3.37 - 3.53i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (3.52 - 4.48i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (5.27 + 3.04i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.820 + 0.947i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.43 + 12.6i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (-0.0847 + 0.211i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (-2.14 + 4.16i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-7.13 - 11.0i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.37 - 3.02i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-11.7 + 4.06i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-5.07 + 4.83i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (2.56 - 17.8i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (7.13 + 1.72i)T + (86.2 + 44.4i)T^{2} \) |
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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
−12.25515340938669002202060569213, −11.34743094238705406492914514653, −9.754985065472950772201664751692, −8.516631731726972964539295328566, −8.115870164259448552581945588703, −6.60079729154067802995176493394, −5.03429278951842324249942040183, −4.85808515888369712362448385500, −2.56151123901915785975934856058, −0.07316399187973143144094468002,
3.58904583852723709667667542238, 4.24014576976224196039626041863, 5.30087773509659786391497715593, 6.81056363555346362293949784996, 7.84779180528155060997438486679, 9.181007096062176784333306644169, 10.35371131419902894895381250946, 10.73291608233174554266591662219, 11.87817329463418781353143707027, 12.90854730381739128003078676747
