L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.837 − 2.07i)5-s + (0.499 − 0.866i)6-s + 0.785i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (1.76 + 1.37i)10-s − 0.377·11-s + 0.999i·12-s + (2.51 + 1.45i)13-s + (−0.392 − 0.680i)14-s + (1.76 + 1.37i)15-s + (−0.5 − 0.866i)16-s + (−2.45 + 1.41i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.374 − 0.927i)5-s + (0.204 − 0.353i)6-s + 0.296i·7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.557 + 0.435i)10-s − 0.113·11-s + 0.288i·12-s + (0.697 + 0.402i)13-s + (−0.104 − 0.181i)14-s + (0.454 + 0.355i)15-s + (−0.125 − 0.216i)16-s + (−0.595 + 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.113283 - 0.242138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.113283 - 0.242138i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.837 + 2.07i)T \) |
| 19 | \( 1 + (2.82 + 3.32i)T \) |
good | 7 | \( 1 - 0.785iT - 7T^{2} \) |
| 11 | \( 1 + 0.377T + 11T^{2} \) |
| 13 | \( 1 + (-2.51 - 1.45i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.45 - 1.41i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (7.86 + 4.54i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.01 + 3.48i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 + 0.0967iT - 37T^{2} \) |
| 41 | \( 1 + (1.43 + 2.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.371 + 0.214i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (10.4 + 6.04i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.00 + 3.46i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.88 + 8.45i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.27 - 3.94i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.31 + 4.22i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.86 - 10.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.95 + 1.13i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.785 + 1.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.75iT - 83T^{2} \) |
| 89 | \( 1 + (-5.96 + 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.98 - 1.72i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.38416091593359195852880671897, −9.394469693780464174155759353493, −8.640147602556204189188073631013, −8.031575390144886085940919775018, −6.69774634458703548557716761506, −5.94801154189764276562417803145, −4.84807166883054307849058768871, −3.96545908697617352225225800763, −1.94415917968314343127077539860, −0.19167503216436158948729416464,
1.74774865446199518683536122956, 3.18784699242236344491500367619, 4.22329899273273051940871231321, 5.85568200115771057093357473372, 6.63464825897014033755328069819, 7.61702674542588835828023720268, 8.211234895420415552990675938813, 9.487666609621544048815017679375, 10.45756449235743937904518487520, 10.89042348659970415740135582784