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 | \( 1 + (-5.53 + 9.59i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.04199781235541659727868287726, −10.05478662105255065397985147300, −9.615543662020809238091045611936, −8.892007810751274972611097673514, −7.79843100735885999652829396207, −6.87254339943428359100315273350, −4.62409026814658935573205881801, −3.23086961078991455579027310810, −2.67306194978508943261679319135, −0.979560271119353689735002071453,
1.99417160528886978841990614605, 4.05800729670641447103988415673, 5.46378152776107744657938531839, 6.72796546557248867585384419342, 7.04484241989644068065594029677, 8.482855032676743090745244987383, 9.117207525325591579360224639941, 9.512058852995104026452357211319, 10.49336042726015830806024234365, 12.24268782236793409036272650000