L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (2.36 − 0.694i)5-s + (0.415 + 0.909i)6-s + (−0.654 − 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−1.61 + 1.86i)10-s + (−2.22 − 1.43i)11-s + (−0.841 − 0.540i)12-s + (−0.672 + 0.776i)13-s + (0.959 + 0.281i)14-s + (−0.350 − 2.43i)15-s + (−0.654 − 0.755i)16-s + (−0.649 − 1.42i)17-s + ⋯ |
L(s) = 1 | + (−0.594 + 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (1.05 − 0.310i)5-s + (0.169 + 0.371i)6-s + (−0.247 − 0.285i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.510 + 0.589i)10-s + (−0.671 − 0.431i)11-s + (−0.242 − 0.156i)12-s + (−0.186 + 0.215i)13-s + (0.256 + 0.0752i)14-s + (−0.0905 − 0.629i)15-s + (−0.163 − 0.188i)16-s + (−0.157 − 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.759015 - 0.851859i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.759015 - 0.851859i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (-2.88 + 3.82i)T \) |
good | 5 | \( 1 + (-2.36 + 0.694i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (2.22 + 1.43i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.672 - 0.776i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.649 + 1.42i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.76 + 6.05i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.11 - 4.62i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.723 + 5.03i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (9.74 + 2.86i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (5.32 - 1.56i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.397 - 2.76i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 8.64T + 47T^{2} \) |
| 53 | \( 1 + (0.976 + 1.12i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.174 - 0.201i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.90 + 13.2i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (0.344 - 0.221i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-7.30 + 4.69i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.03 + 2.26i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (7.23 - 8.34i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-14.5 - 4.28i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.22 + 8.54i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (4.06 - 1.19i)T + (81.6 - 52.4i)T^{2} \) |
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
−9.532610376999373307347315753785, −9.047188939779029398252994225569, −8.176455317939609088533590775464, −7.13897110631272898025324927460, −6.61986176834048187033934410023, −5.56550661286382019344081691887, −4.88954932611025089743280471997, −3.03517296711656921132623071617, −2.00190512483848022782037728449, −0.61906035603042443556578733331,
1.70745801223789330564027111599, 2.73446536406558917367457094523, 3.70907992714266008402920946090, 5.15758609709516178486958910310, 5.83953101887657030944332386344, 6.93595540550480194442195509561, 7.907939082490656973912343162408, 8.827157686966060386246755468952, 9.585823536788306397194306552215, 10.38454780479964561962304975388