L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·6-s + (1.89 − 3.28i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 3.84·11-s + (0.499 + 0.866i)12-s + (5.82 + 3.36i)13-s − 3.78i·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (4.14 − 2.39i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 0.408i·6-s + (0.715 − 1.23i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 1.15·11-s + (0.144 + 0.249i)12-s + (1.61 + 0.932i)13-s − 1.01i·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (1.00 − 0.580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.386887441\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.386887441\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 37 | \( 1 + (-2.29 - 5.63i)T \) |
good | 7 | \( 1 + (-1.89 + 3.28i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 + (-5.82 - 3.36i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.14 + 2.39i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.79 + 2.76i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 8.45iT - 23T^{2} \) |
| 29 | \( 1 + 1.86iT - 29T^{2} \) |
| 31 | \( 1 - 8.49iT - 31T^{2} \) |
| 41 | \( 1 + (-4.99 + 8.65i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 0.0397iT - 43T^{2} \) |
| 47 | \( 1 - 1.04T + 47T^{2} \) |
| 53 | \( 1 + (1.60 + 2.77i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.56 + 0.902i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.4 - 6.62i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.58 + 7.94i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.928 - 1.60i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + (3.13 + 1.80i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.79 - 11.7i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-14.8 + 8.55i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.15iT - 97T^{2} \) |
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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
−10.23186233856283226071653385599, −8.961338688969667570448279837593, −8.134184902704045985470206587695, −6.97103159953830113831567470789, −6.29581452944118063071701471638, −5.19255265144755295424748908991, −4.48153156468493188952049907565, −3.68493312002592554316942893608, −2.44104215565122139786347283566, −0.993130458707182892315127290757,
1.53124206254629286973398831867, 2.61507549061848882803960574565, 3.82223506617132766537714106131, 5.22342299306108722001179183540, 5.79628153173171957772622267567, 6.06614527237441064927399429582, 7.74917204100502922246451680228, 8.030789611724284809357914835955, 8.836149669904354763092702701367, 10.05623968730752048799031865523