| L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.0875 + 0.608i)5-s + (0.841 + 0.540i)6-s + (0.415 − 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.255 − 0.559i)10-s + (−2.70 − 0.795i)11-s + (−0.959 − 0.281i)12-s + (1.87 + 4.11i)13-s + (−0.142 + 0.989i)14-s + (0.402 − 0.464i)15-s + (0.415 − 0.909i)16-s + (−1.98 − 1.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (−0.378 − 0.436i)3-s + (0.420 − 0.270i)4-s + (0.0391 + 0.272i)5-s + (0.343 + 0.220i)6-s + (0.157 − 0.343i)7-s + (−0.231 + 0.267i)8-s + (−0.0474 + 0.329i)9-s + (−0.0808 − 0.176i)10-s + (−0.816 − 0.239i)11-s + (−0.276 − 0.0813i)12-s + (0.520 + 1.14i)13-s + (−0.0380 + 0.264i)14-s + (0.104 − 0.120i)15-s + (0.103 − 0.227i)16-s + (−0.480 − 0.308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.206740 + 0.373236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.206740 + 0.373236i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (4.61 + 1.30i)T \) |
| good | 5 | \( 1 + (-0.0875 - 0.608i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (2.70 + 0.795i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.87 - 4.11i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (1.98 + 1.27i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.17 - 2.03i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 0.718i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (4.39 - 5.07i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.477 - 3.32i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.0958 - 0.666i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.753 - 0.869i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 2.26T + 47T^{2} \) |
| 53 | \( 1 + (3.79 - 8.31i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.414 - 0.907i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.37 + 1.58i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (6.22 - 1.82i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (11.4 - 3.36i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-4.88 + 3.13i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-6.37 - 13.9i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.88 - 13.0i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.273 - 0.315i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.41 - 16.8i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.58579589745618403931008366320, −9.400080356616491254914885449915, −8.521337622369402216642003826170, −7.84764415006744926898584010405, −6.82363180269816579124181883718, −6.38825865488512507547098598475, −5.26266918930550239401147429059, −4.14190397721868892934332793977, −2.62696932371648989888737574396, −1.46201673404043313661514550291,
0.25701224261657893480056811342, 1.99106855871762862091344409791, 3.20478707969387160645262275309, 4.43426716826177478857883816701, 5.46372657482731656532805371361, 6.21262407321273257582634521633, 7.41304634055227884555298957075, 8.256897105260741748768134095342, 8.901347048094428988069817475302, 9.816826079846093898779680677812
