L(s) = 1 | + (−3.99 − 6.93i)2-s + (31.6 − 31.6i)3-s + (−32.1 + 55.3i)4-s + (69.9 − 69.9i)5-s + (−346. − 93.0i)6-s − 445.·7-s + (511. + 1.51i)8-s − 1.27e3i·9-s + (−764. − 205. i)10-s + (365. + 365. i)11-s + (736. + 2.77e3i)12-s + (1.78e3 + 1.78e3i)13-s + (1.77e3 + 3.08e3i)14-s − 4.43e3i·15-s + (−2.03e3 − 3.55e3i)16-s + 2.87e3·17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.866i)2-s + (1.17 − 1.17i)3-s + (−0.501 + 0.865i)4-s + (0.559 − 0.559i)5-s + (−1.60 − 0.431i)6-s − 1.29·7-s + (0.999 + 0.00294i)8-s − 1.75i·9-s + (−0.764 − 0.205i)10-s + (0.274 + 0.274i)11-s + (0.426 + 1.60i)12-s + (0.813 + 0.813i)13-s + (0.647 + 1.12i)14-s − 1.31i·15-s + (−0.496 − 0.867i)16-s + 0.585·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.790i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.611 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.663431 - 1.35200i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.663431 - 1.35200i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.99 + 6.93i)T \) |
good | 3 | \( 1 + (-31.6 + 31.6i)T - 729iT^{2} \) |
| 5 | \( 1 + (-69.9 + 69.9i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 + 445.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-365. - 365. i)T + 1.77e6iT^{2} \) |
| 13 | \( 1 + (-1.78e3 - 1.78e3i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 - 2.87e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-8.63e3 + 8.63e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 + 8.37e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.38e4 - 1.38e4i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 - 2.06e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (3.70e3 - 3.70e3i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 - 4.58e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (1.00e5 + 1.00e5i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 - 3.67e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (8.38e4 - 8.38e4i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 + (-4.31e4 - 4.31e4i)T + 4.21e10iT^{2} \) |
| 61 | \( 1 + (-7.90e4 - 7.90e4i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 + (1.51e5 - 1.51e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 + 3.83e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 1.01e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 3.80e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-4.39e4 + 4.39e4i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 - 6.31e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.05e6T + 8.32e11T^{2} \) |
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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
−17.85268703461581126287136214432, −16.26680253360406893694522834196, −13.83593784812234085920870899009, −13.20735053823140760336966844848, −12.08323177280129973040015689611, −9.622199371024602151864056583255, −8.766344861156987933666870948798, −7.00233420400668316031654740211, −3.15229677349882731285340120476, −1.35911019554165587882522998169,
3.43671713636622691714872819785, 6.01343910185102858702815457736, 8.148126435254602651269155218249, 9.688143527961812345048531110813, 10.17745011738480400585342741935, 13.57941403250593474246754547584, 14.46764357393952309431910695914, 15.73387815056333862267718059582, 16.42192460669580233264556404651, 18.28697487572840275447959538167