L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.500i)10-s + (−1.86 + 0.5i)13-s + (0.500 + 0.866i)16-s + (−1.22 − 1.22i)17-s + (0.258 + 0.965i)20-s + 1.00i·25-s + 1.93i·26-s + (−0.258 − 0.448i)29-s + (0.965 − 0.258i)32-s + (−1.49 + 0.866i)34-s + (−0.366 + 0.366i)37-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.500i)10-s + (−1.86 + 0.5i)13-s + (0.500 + 0.866i)16-s + (−1.22 − 1.22i)17-s + (0.258 + 0.965i)20-s + 1.00i·25-s + 1.93i·26-s + (−0.258 − 0.448i)29-s + (0.965 − 0.258i)32-s + (−1.49 + 0.866i)34-s + (−0.366 + 0.366i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3266744982\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3266744982\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 41 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - 1.93T + T^{2} \) |
| 97 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.303299817422522918880037289845, −8.526432867380156776785225621680, −7.55499331775721873161374304488, −6.74725739927049135160231634257, −5.29346475573320965160002980629, −4.77829918804097446720769971987, −4.09112577556449286261716250049, −2.88862077082545931475789721635, −1.94105871708479458431107976979, −0.22036881706556110741650052589,
2.44074904524439490058349553075, 3.51370806710728590956459472029, 4.39884896669995591127354918611, 5.17429122411093693226773662989, 6.24008615606684594982405912378, 6.98003531874669966857616003131, 7.54685370387431367707473851615, 8.295597723539941944845491282380, 9.073572158735422179946617579332, 10.08660713876941276436094716277