| L(s) = 1 | + (−0.365 − 0.930i)2-s + (0.179 + 2.39i)3-s + (−0.733 + 0.680i)4-s + (−3.27 + 2.23i)5-s + (2.16 − 1.04i)6-s + (0.900 + 0.433i)8-s + (−2.73 + 0.411i)9-s + (3.27 + 2.23i)10-s + (−3.91 − 0.590i)11-s + (−1.75 − 1.63i)12-s + (−1.13 + 1.42i)13-s + (−5.92 − 7.43i)15-s + (0.0747 − 0.997i)16-s + (1.24 − 0.382i)17-s + (1.38 + 2.39i)18-s + (1.57 − 2.73i)19-s + ⋯ |
| L(s) = 1 | + (−0.258 − 0.658i)2-s + (0.103 + 1.38i)3-s + (−0.366 + 0.340i)4-s + (−1.46 + 0.997i)5-s + (0.882 − 0.425i)6-s + (0.318 + 0.153i)8-s + (−0.910 + 0.137i)9-s + (1.03 + 0.705i)10-s + (−1.18 − 0.177i)11-s + (−0.507 − 0.471i)12-s + (−0.314 + 0.394i)13-s + (−1.52 − 1.91i)15-s + (0.0186 − 0.249i)16-s + (0.300 − 0.0928i)17-s + (0.325 + 0.563i)18-s + (0.361 − 0.626i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 + 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0524690 - 0.133077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0524690 - 0.133077i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.365 + 0.930i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-0.179 - 2.39i)T + (-2.96 + 0.447i)T^{2} \) |
| 5 | \( 1 + (3.27 - 2.23i)T + (1.82 - 4.65i)T^{2} \) |
| 11 | \( 1 + (3.91 + 0.590i)T + (10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (1.13 - 1.42i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.24 + 0.382i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-1.57 + 2.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.96 - 0.606i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-1.60 + 7.04i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.97 - 3.41i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.08 - 1.00i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (8.00 + 3.85i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (5.80 - 2.79i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (2.39 + 6.10i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (7.48 - 6.94i)T + (3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (11.1 + 7.60i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (-4.15 - 3.85i)T + (4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.482 - 0.835i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.18 - 5.21i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (1.49 - 3.80i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-6.03 + 10.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.791 - 0.992i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (12.7 - 1.92i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.89915034623531667486300016808, −10.28172483447960734722082230084, −9.606112745102418126685803270196, −8.482247412473001862496900195329, −7.78591861225528515104036674609, −6.82908363446362721542163002068, −5.11292350233920677668913841246, −4.36405303054724801395490185381, −3.38818479036076448020758159469, −2.79121712236017766889379899510,
0.086584316649192647770582541413, 1.37402153425356712302305545835, 3.17271651007355432459198819633, 4.65493121762786748165819834110, 5.43057101318567201028280344488, 6.73770395022669659872864453802, 7.55316813140120166079922868887, 8.039186557300159949172610546338, 8.481604202381671970552983579447, 9.740593030932481764340060550829
