L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (1 − 1.73i)9-s + (0.5 + 0.866i)11-s − 5·13-s − 0.999·15-s + (−0.5 − 0.866i)17-s + (3 − 5.19i)19-s + (2 − 3.46i)23-s + (−0.499 − 0.866i)25-s − 5·27-s + 3·29-s + (−1 − 1.73i)31-s + (0.499 − 0.866i)33-s + (−4 + 6.92i)37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.333 − 0.577i)9-s + (0.150 + 0.261i)11-s − 1.38·13-s − 0.258·15-s + (−0.121 − 0.210i)17-s + (0.688 − 1.19i)19-s + (0.417 − 0.722i)23-s + (−0.0999 − 0.173i)25-s − 0.962·27-s + 0.557·29-s + (−0.179 − 0.311i)31-s + (0.0870 − 0.150i)33-s + (−0.657 + 1.13i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.501100 - 1.01083i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.501100 - 1.01083i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3T + 97T^{2} \) |
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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
−9.637486934578038722909565588879, −9.028811615725359930153200547915, −7.938528198594060180773560421516, −6.95375780348520040040816799392, −6.58160469481264231171624965293, −5.19639133130327846629235103472, −4.65920022162677864775852065666, −3.20342815647747093083259013913, −1.95487639687081214359041701397, −0.53186778408631711985954219116,
1.73665222995988326943602610498, 3.02144290062743526287598137725, 4.14208950539746391445099325468, 5.15465273900548876370272694088, 5.79571053389078322178471225137, 7.07597207526028365304325457043, 7.60335019916407054816027280971, 8.725715283939267003927044676210, 9.765994721157904116722209607402, 10.18532959482445216862770727769