L(s) = 1 | + (−0.395 + 0.228i)2-s + (−0.866 + 0.5i)3-s + (−0.895 + 1.55i)4-s + (0.456 − 2.18i)5-s + (0.228 − 0.395i)6-s + (1.5 + 0.866i)7-s − 1.73i·8-s + (−1 + 1.73i)9-s + (0.319 + 0.970i)10-s + (−1.32 − 2.29i)11-s − 1.79i·12-s − 0.791·14-s + (0.698 + 2.12i)15-s + (−1.39 − 2.41i)16-s + (−3.96 − 2.29i)17-s − 0.913i·18-s + ⋯ |
L(s) = 1 | + (−0.279 + 0.161i)2-s + (−0.499 + 0.288i)3-s + (−0.447 + 0.775i)4-s + (0.204 − 0.978i)5-s + (0.0932 − 0.161i)6-s + (0.566 + 0.327i)7-s − 0.612i·8-s + (−0.333 + 0.577i)9-s + (0.100 + 0.306i)10-s + (−0.398 − 0.690i)11-s − 0.517i·12-s − 0.211·14-s + (0.180 + 0.548i)15-s + (−0.348 − 0.604i)16-s + (−0.962 − 0.555i)17-s − 0.215i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.398182 - 0.350622i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.398182 - 0.350622i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.456 + 2.18i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.395 - 0.228i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.32 + 2.29i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.96 + 2.29i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.866 - 1.5i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.29 - 3.96i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.20T + 31T^{2} \) |
| 37 | \( 1 + (-6.87 + 3.96i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.32 + 2.29i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (9.16 + 5.29i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 1.82iT - 47T^{2} \) |
| 53 | \( 1 + 7.58iT - 53T^{2} \) |
| 59 | \( 1 + (-6.97 + 12.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.708 + 1.22i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.873 + 0.504i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.51 + 6.08i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 6.01iT - 83T^{2} \) |
| 89 | \( 1 + (-4.78 - 8.29i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.87 + 5.70i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.867869363034313699672645808214, −8.844048134860861537359864916891, −8.520856931245462637500891984226, −7.72620417668314555825646868398, −6.52264403037297077428735373339, −5.12432132898003682020329852402, −5.01981065276842365057349590190, −3.72285679771165619541033232767, −2.25006629265932881855781520884, −0.31611957890291589599277168241,
1.39574610462954580618759190041, 2.64337005344953564427304925575, 4.18753356410278346018543387234, 5.17503618106191266131859650561, 6.14235634758657530039025841705, 6.78948284460291727600767386554, 7.78405149535495674245803923299, 8.870307712993754595566873664492, 9.689473478334734338985186797860, 10.44447239031014411694084690215