L(s) = 1 | + (−1.29 + 2.23i)2-s + (1.72 + 0.177i)3-s + (−2.34 − 4.05i)4-s + 5-s + (−2.62 + 3.62i)6-s + (−2.13 + 1.55i)7-s + 6.94·8-s + (2.93 + 0.612i)9-s + (−1.29 + 2.23i)10-s + 2.47·11-s + (−3.31 − 7.40i)12-s + (−3.19 + 5.52i)13-s + (−0.718 − 6.80i)14-s + (1.72 + 0.177i)15-s + (−4.29 + 7.43i)16-s + (−2.57 + 4.45i)17-s + ⋯ |
L(s) = 1 | + (−0.914 + 1.58i)2-s + (0.994 + 0.102i)3-s + (−1.17 − 2.02i)4-s + 0.447·5-s + (−1.07 + 1.48i)6-s + (−0.808 + 0.588i)7-s + 2.45·8-s + (0.978 + 0.204i)9-s + (−0.408 + 0.708i)10-s + 0.744·11-s + (−0.957 − 2.13i)12-s + (−0.885 + 1.53i)13-s + (−0.191 − 1.81i)14-s + (0.444 + 0.0458i)15-s + (−1.07 + 1.85i)16-s + (−0.623 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.264814 + 1.01028i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.264814 + 1.01028i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.72 - 0.177i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.13 - 1.55i)T \) |
good | 2 | \( 1 + (1.29 - 2.23i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + (3.19 - 5.52i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.57 - 4.45i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.825 - 1.42i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 + (-0.321 - 0.557i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.273 + 0.474i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.30 + 5.72i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.42 + 7.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.75 + 6.50i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.817 + 1.41i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.66 + 8.07i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.44 - 7.70i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.157 + 0.272i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.593 - 1.02i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 + (-6.30 + 10.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.14 - 10.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.21 - 5.57i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.44 + 2.50i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.53 - 9.59i)T + (-48.5 + 84.0i)T^{2} \) |
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
−12.24268782236793409036272650000, −10.49336042726015830806024234365, −9.512058852995104026452357211319, −9.117207525325591579360224639941, −8.482855032676743090745244987383, −7.04484241989644068065594029677, −6.72796546557248867585384419342, −5.46378152776107744657938531839, −4.05800729670641447103988415673, −1.99417160528886978841990614605,
0.979560271119353689735002071453, 2.67306194978508943261679319135, 3.23086961078991455579027310810, 4.62409026814658935573205881801, 6.87254339943428359100315273350, 7.79843100735885999652829396207, 8.892007810751274972611097673514, 9.615543662020809238091045611936, 10.05478662105255065397985147300, 11.04199781235541659727868287726