L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.59 − 2.75i)5-s + (1.85 + 1.88i)7-s + 0.999·8-s + (−1.59 + 2.75i)10-s + (1.59 − 2.75i)11-s − 5.70·13-s + (0.710 − 2.54i)14-s + (−0.5 − 0.866i)16-s + (0.760 − 1.31i)17-s + (−0.641 − 1.11i)19-s + 3.18·20-s − 3.18·22-s + (1.11 + 1.93i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.711 − 1.23i)5-s + (0.699 + 0.714i)7-s + 0.353·8-s + (−0.503 + 0.871i)10-s + (0.479 − 0.830i)11-s − 1.58·13-s + (0.189 − 0.681i)14-s + (−0.125 − 0.216i)16-s + (0.184 − 0.319i)17-s + (−0.147 − 0.254i)19-s + 0.711·20-s − 0.678·22-s + (0.233 + 0.404i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5055487909\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5055487909\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.85 - 1.88i)T \) |
good | 5 | \( 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.59 + 2.75i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 + (-0.760 + 1.31i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.11 - 1.93i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 + (-4.71 + 8.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.02 - 1.78i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.562 - 0.974i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.56 + 2.70i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.69T + 71T^{2} \) |
| 73 | \( 1 + (2.48 - 4.30i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8.06T + 83T^{2} \) |
| 89 | \( 1 + (0.112 + 0.195i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−9.304822237490832272070226200858, −8.591330438333720451352860962273, −8.028064511971727839008011143410, −7.19310403338197680015007828332, −5.65955907743716892217633638594, −4.89972695051314991335802751009, −4.14180611614753255039484140957, −2.86524750312932327889793592580, −1.62538141777915267644778850861, −0.24859858381071957942222525630,
1.77996875609968237665201028803, 3.20104012483558732629234410214, 4.33750357077572306333313510682, 5.05314678469189144969487658657, 6.48426727416012360372871504260, 7.15286716769967787212514989838, 7.54172922146953703830459216291, 8.369648717504189802660153443200, 9.560175934747778519703867113863, 10.26997909054421135601695241714