L(s) = 1 | + (0.366 + 0.633i)2-s + (0.732 − 1.26i)4-s + (0.5 + 0.866i)5-s + (−0.866 + 2.5i)7-s + 2.53·8-s + (−0.366 + 0.633i)10-s + (−1.36 + 2.36i)11-s + 5.73·13-s + (−1.90 + 0.366i)14-s + (−0.535 − 0.928i)16-s + (3.36 − 5.83i)17-s + (1.23 + 2.13i)19-s + 1.46·20-s − 2·22-s + (−0.633 − 1.09i)23-s + ⋯ |
L(s) = 1 | + (0.258 + 0.448i)2-s + (0.366 − 0.633i)4-s + (0.223 + 0.387i)5-s + (−0.327 + 0.944i)7-s + 0.896·8-s + (−0.115 + 0.200i)10-s + (−0.411 + 0.713i)11-s + 1.58·13-s + (−0.508 + 0.0978i)14-s + (−0.133 − 0.232i)16-s + (0.816 − 1.41i)17-s + (0.282 + 0.489i)19-s + 0.327·20-s − 0.426·22-s + (−0.132 − 0.228i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.63550 + 0.551397i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63550 + 0.551397i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.866 - 2.5i)T \) |
good | 2 | \( 1 + (-0.366 - 0.633i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 + (-3.36 + 5.83i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.23 - 2.13i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.633 + 1.09i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 + (3.23 - 5.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.59 + 6.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 + 7.19T + 43T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.19 - 7.26i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.09 + 8.83i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.33 - 2.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 + (-2.33 + 4.03i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.69 + 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + (4.56 + 7.90i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−11.68900656525137058385025897230, −10.81400802362530962464848811775, −9.911397818818659087827763038972, −9.048867729666445984504518058966, −7.70706358175997892451873797249, −6.75437665933937998662421897053, −5.80560887138700782961388707185, −5.12005318141173849454984882081, −3.32749106160138390791360595062, −1.83928850010978849005790941824,
1.48615949979414703311502517711, 3.36641935276834435815412929471, 3.96687288362390057868290466107, 5.59934437717660654300007051289, 6.68581478628377410754662477614, 7.889191619018634660923270362022, 8.543032861985796281315126504284, 9.957309355197011839922065963071, 10.84447166400918347428182933881, 11.42410248507747452156652041823