L(s) = 1 | + (−1.73 − i)3-s + (1.73 + 2i)7-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + 3i·13-s + (1.73 + i)17-s + (−2.5 − 4.33i)19-s + (−0.999 − 5.19i)21-s + (−6.06 + 3.5i)23-s + 4.00i·27-s + 6·29-s + (−2 + 3.46i)31-s + (−1.73 + 0.999i)33-s + (−4.33 + 2.5i)37-s + (3 − 5.19i)39-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.577i)3-s + (0.654 + 0.755i)7-s + (0.166 + 0.288i)9-s + (0.150 − 0.261i)11-s + 0.832i·13-s + (0.420 + 0.242i)17-s + (−0.573 − 0.993i)19-s + (−0.218 − 1.13i)21-s + (−1.26 + 0.729i)23-s + 0.769i·27-s + 1.11·29-s + (−0.359 + 0.622i)31-s + (−0.301 + 0.174i)33-s + (−0.711 + 0.410i)37-s + (0.480 − 0.832i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8571657945\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8571657945\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 3 | \( 1 + (1.73 + i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.06 - 3.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.33 - 2.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + (7.79 - 4.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.52 - 5.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 - 2i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (-10.3 - 6i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.727641460871687400619742156702, −8.773500362635987673891080849186, −8.203476836995063119300897092557, −7.01411197030077772484136897056, −6.47887112433296232073742489007, −5.58670142686039896710079978726, −4.96541716030442589235518553448, −3.78452168032819384951494684738, −2.33219141980804770340483815664, −1.25058585158651339437805787567,
0.43085976744882913828925056628, 1.97601280831452375196762100091, 3.58395363244916475216193540501, 4.44006574932971259656464994554, 5.17153634662819944737670411144, 5.96972413819591610267129714264, 6.82708818289721519086904105408, 7.964790814646868141561078343263, 8.347861814874528299313792382454, 9.853544576748644376492461665358