L(s) = 1 | + (0.866 − 1.5i)3-s + (−1.73 − 2i)7-s + (−1.5 − 2.59i)9-s + (−4.5 + 2.59i)11-s + (2.59 − 1.5i)17-s + (−1.5 − 0.866i)19-s + (−4.5 + 0.866i)21-s + (−2.59 + 4.5i)23-s − 5.19·27-s + (−1.5 + 0.866i)31-s + 9i·33-s + (6.06 + 3.5i)37-s − 6·41-s + 4i·43-s + (−2.59 − 1.5i)47-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)3-s + (−0.654 − 0.755i)7-s + (−0.5 − 0.866i)9-s + (−1.35 + 0.783i)11-s + (0.630 − 0.363i)17-s + (−0.344 − 0.198i)19-s + (−0.981 + 0.188i)21-s + (−0.541 + 0.938i)23-s − 1.00·27-s + (−0.269 + 0.155i)31-s + 1.56i·33-s + (0.996 + 0.575i)37-s − 0.937·41-s + 0.609i·43-s + (−0.378 − 0.218i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 + 2i)T \) |
good | 11 | \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-2.59 + 1.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 - 4.5i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.06 - 3.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (2.59 + 1.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.59 - 4.5i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.5 + 6.06i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.33 - 2.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (6.06 + 10.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.376739009620499584768471272891, −7.64380018155060398732130642266, −7.26417974233690903751297434802, −6.38485460835652890825387728369, −5.51377896887115323625019517544, −4.44606746361621602419797657604, −3.34297855185299491499346568796, −2.61787826604039385373127406938, −1.43744347115331813429196001157, 0,
2.22114959563591971743055697786, 2.96883282160106709929240573244, 3.74869272563242552627648138856, 4.84042675216014226789944528880, 5.64359965940695712022344153467, 6.20938857716596595247317117082, 7.55885076103154126001517862205, 8.284339404353689536782466728406, 8.774454367406357026911727915995