L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (0.279 − 0.0819i)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 + (−0.190 + 0.219i)10-s + (1.72 + 1.10i)11-s + (−0.841 − 0.540i)12-s + (2.81 − 3.24i)13-s + (−0.959 − 0.281i)14-s + (−0.0414 − 0.288i)15-s + (−0.654 − 0.755i)16-s + (−0.928 − 2.03i)17-s + ⋯ |
L(s) = 1 | + (−0.594 + 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (0.124 − 0.0366i)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 + (−0.0602 + 0.0695i)10-s + (0.520 + 0.334i)11-s + (−0.242 − 0.156i)12-s + (0.779 − 0.900i)13-s + (−0.256 − 0.0752i)14-s + (−0.0106 − 0.0743i)15-s + (−0.163 − 0.188i)16-s + (−0.225 − 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28873 - 0.325181i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28873 - 0.325181i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.48 - 4.09i)T \) |
good | 5 | \( 1 + (-0.279 + 0.0819i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (-1.72 - 1.10i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.81 + 3.24i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.928 + 2.03i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.997 + 2.18i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.60 - 3.52i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.775 + 5.39i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-9.76 - 2.86i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-8.76 + 2.57i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.756 + 5.26i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 3.78T + 47T^{2} \) |
| 53 | \( 1 + (5.29 + 6.10i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (5.68 - 6.56i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.45 + 10.1i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-13.2 + 8.54i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-10.1 + 6.49i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.26 + 9.33i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.338 - 0.390i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.61 - 1.64i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.273 + 1.90i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (15.9 - 4.67i)T + (81.6 - 52.4i)T^{2} \) |
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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
−9.525963984275333579381191892642, −9.254139327923607910704108943663, −8.097247793844320618489804994268, −7.61658029607277569688385016885, −6.62357145494709783667484193722, −5.85088390933317800744962730058, −4.94878455631989263244235353348, −3.45564528111463099345124088332, −2.15603516152482335177696536359, −0.915776674066667046530883090342,
1.21329735053027337723905380878, 2.56903527319362423744660581003, 3.85440520853365034685368936336, 4.44547900546166976768628649990, 5.94759945037974701531304543187, 6.64663508087105493826589757924, 7.896655285887447238648695179375, 8.535237328311531069497525843910, 9.368968498008747309230900237861, 9.965023783869496039421812029968