| L(s) = 1 | + (0.0727 − 1.41i)2-s + (−0.608 + 0.793i)3-s + (−1.98 − 0.205i)4-s + (−0.419 − 0.547i)5-s + (1.07 + 0.917i)6-s + (0.946 + 2.47i)7-s + (−0.434 + 2.79i)8-s + (−0.258 − 0.965i)9-s + (−0.803 + 0.553i)10-s + (−0.630 − 4.78i)11-s + (1.37 − 1.45i)12-s + (0.146 + 0.353i)13-s + (3.55 − 1.15i)14-s + 0.689·15-s + (3.91 + 0.817i)16-s + (−0.687 − 1.19i)17-s + ⋯ |
| L(s) = 1 | + (0.0514 − 0.998i)2-s + (−0.351 + 0.458i)3-s + (−0.994 − 0.102i)4-s + (−0.187 − 0.244i)5-s + (0.439 + 0.374i)6-s + (0.357 + 0.933i)7-s + (−0.153 + 0.988i)8-s + (−0.0862 − 0.321i)9-s + (−0.254 + 0.174i)10-s + (−0.189 − 1.44i)11-s + (0.396 − 0.419i)12-s + (0.0406 + 0.0981i)13-s + (0.950 − 0.309i)14-s + 0.178·15-s + (0.978 + 0.204i)16-s + (−0.166 − 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.531163 - 0.871566i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.531163 - 0.871566i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0727 + 1.41i)T \) |
| 3 | \( 1 + (0.608 - 0.793i)T \) |
| 7 | \( 1 + (-0.946 - 2.47i)T \) |
| good | 5 | \( 1 + (0.419 + 0.547i)T + (-1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (0.630 + 4.78i)T + (-10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (-0.146 - 0.353i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (0.687 + 1.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.435 + 3.31i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (-7.18 + 1.92i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.61 + 6.31i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (-0.758 - 1.31i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.558 - 0.428i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (7.26 + 7.26i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.63 - 1.09i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.78 - 1.03i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.03 - 0.268i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (-0.277 - 2.11i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.888 + 6.74i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (4.59 + 3.52i)T + (17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-8.35 + 8.35i)T - 71iT^{2} \) |
| 73 | \( 1 + (0.998 - 3.72i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.987 + 1.70i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.9 + 4.53i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-3.88 - 14.4i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 - 14.8iT - 97T^{2} \) |
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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
−10.51527675418165401063795082876, −9.260054936556725942416050648283, −8.848346810120393665024378439216, −8.029395441754088038747472945257, −6.35000461933723636486303910381, −5.33810745668152395733442662087, −4.70922925016513534914744665612, −3.43748187350065354097485886678, −2.45753840248684480159138921303, −0.62652679220246292673278431529,
1.40626827095198788337115654472, 3.48529212047189531610590203693, 4.63479510211074991775672272147, 5.35093783722077177546491519129, 6.65182360272778144144561625339, 7.25810748704539214516793320100, 7.74570162778453774302301910869, 8.875361671695624608100298192574, 9.911553963462937331336619835648, 10.65618049538824160068592850918
