L(s) = 1 | + 1.41i·2-s − 1.00·4-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)13-s + (1.22 + 0.707i)14-s − 0.999·16-s + 1.41i·17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (−1.22 + 0.707i)26-s + (−0.500 + 0.866i)28-s + (1.22 − 0.707i)29-s − 1.41i·32-s − 2.00·34-s − 37-s + (−1.22 + 0.707i)38-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 1.00·4-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)13-s + (1.22 + 0.707i)14-s − 0.999·16-s + 1.41i·17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (−1.22 + 0.707i)26-s + (−0.500 + 0.866i)28-s + (1.22 − 0.707i)29-s − 1.41i·32-s − 2.00·34-s − 37-s + (−1.22 + 0.707i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.029783623\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.029783623\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.57221404271902931027929551787, −9.781814032085503847713707512844, −8.400248797839034787321723188066, −8.258791976304104379901564714450, −7.17377937799014902371952014654, −6.47433388570937445379467678298, −5.69729838169400291796311246324, −4.56723456385672906717109969695, −3.79948890656025790740259195610, −1.80193875955297551872612182684,
1.26642902198179722273586619182, 2.63935265313141468885693897318, 3.24899284705905564046074302253, 4.67845296221088674139930790286, 5.41428046205412410644214536467, 6.71744815663180844223210719724, 7.80159259005987348873172886695, 8.880467252971510345294615877519, 9.419745271965381845394299387195, 10.34704394975137153386494664312