L(s) = 1 | + (−0.995 − 0.0950i)2-s + (0.396 + 0.254i)3-s + (0.981 + 0.189i)4-s + (−0.370 − 0.291i)6-s + (−0.959 − 0.281i)8-s + (−0.323 − 0.707i)9-s + (1.50 − 1.18i)11-s + (0.341 + 0.325i)12-s + (0.928 + 0.371i)16-s + (−0.642 + 0.123i)17-s + (0.254 + 0.734i)18-s + (1.07 + 1.51i)19-s + (−1.61 + 1.03i)22-s + (−0.308 − 0.356i)24-s + (−0.959 + 0.281i)25-s + ⋯ |
L(s) = 1 | + (−0.995 − 0.0950i)2-s + (0.396 + 0.254i)3-s + (0.981 + 0.189i)4-s + (−0.370 − 0.291i)6-s + (−0.959 − 0.281i)8-s + (−0.323 − 0.707i)9-s + (1.50 − 1.18i)11-s + (0.341 + 0.325i)12-s + (0.928 + 0.371i)16-s + (−0.642 + 0.123i)17-s + (0.254 + 0.734i)18-s + (1.07 + 1.51i)19-s + (−1.61 + 1.03i)22-s + (−0.308 − 0.356i)24-s + (−0.959 + 0.281i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6948736110\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6948736110\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.995 + 0.0950i)T \) |
| 67 | \( 1 + (0.142 + 0.989i)T \) |
good | 3 | \( 1 + (-0.396 - 0.254i)T + (0.415 + 0.909i)T^{2} \) |
| 5 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 7 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 11 | \( 1 + (-1.50 + 1.18i)T + (0.235 - 0.971i)T^{2} \) |
| 13 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 17 | \( 1 + (0.642 - 0.123i)T + (0.928 - 0.371i)T^{2} \) |
| 19 | \( 1 + (-1.07 - 1.51i)T + (-0.327 + 0.945i)T^{2} \) |
| 23 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.271 - 0.785i)T + (-0.786 - 0.618i)T^{2} \) |
| 43 | \( 1 + (-1.02 - 1.18i)T + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (1.38 + 0.407i)T + (0.841 + 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 71 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 73 | \( 1 + (1.54 + 1.21i)T + (0.235 + 0.971i)T^{2} \) |
| 79 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 83 | \( 1 + (0.264 + 0.105i)T + (0.723 + 0.690i)T^{2} \) |
| 89 | \( 1 + (1.67 - 1.07i)T + (0.415 - 0.909i)T^{2} \) |
| 97 | \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−11.01875671426376306739607797861, −9.792060919711475165429666605232, −9.307128714684571693800387585777, −8.526244120843584957118020588682, −7.73466062822486655281106521340, −6.43304232167198870424123805398, −5.92073980492392829922034946796, −3.87400967809046699814197577673, −3.15146009486399960560981690455, −1.40486506447869574628400611934,
1.67418817551595387719434100082, 2.76456282467137060877622230801, 4.39091554942232282356443813882, 5.76027820241299323055786046252, 7.06854714196436060699481830370, 7.31439477935093135093062017042, 8.601581825872844664329934819142, 9.202864280248118305688536910518, 9.933834800421357435750268852147, 11.07080051734138537717900017158