L(s) = 1 | + (−0.5 + 0.866i)3-s − 5-s + (0.5 + 0.866i)7-s + (1 + 1.73i)9-s + (−1.5 + 2.59i)11-s + (1 + 3.46i)13-s + (0.5 − 0.866i)15-s + (1.5 + 2.59i)17-s + (−2.5 − 4.33i)19-s − 0.999·21-s + (−4.5 + 7.79i)23-s + 25-s − 5·27-s + (4.5 − 7.79i)29-s + 8·31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s − 0.447·5-s + (0.188 + 0.327i)7-s + (0.333 + 0.577i)9-s + (−0.452 + 0.783i)11-s + (0.277 + 0.960i)13-s + (0.129 − 0.223i)15-s + (0.363 + 0.630i)17-s + (−0.573 − 0.993i)19-s − 0.218·21-s + (−0.938 + 1.62i)23-s + 0.200·25-s − 0.962·27-s + (0.835 − 1.44i)29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.704367 + 0.713458i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.704367 + 0.713458i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (-1 - 3.46i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.5 + 7.79i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.98305682037253463131741967335, −11.35753933106026292011482398084, −10.30509448922771641790855168684, −9.545910646661125405353337508357, −8.282733700546634353088809613062, −7.42147685506373683673133857360, −6.12218659300924945656328048533, −4.83472529881436202340352022996, −4.05170959812265054789947377536, −2.15027365314411262605981248312,
0.856557334826550739761263451070, 3.03568569313595587159094219180, 4.37497526960047177461463069333, 5.80750831831955318944786424904, 6.72667613616255717497304742582, 7.907038550521635541879595245296, 8.547322484348089346348842264120, 10.12987672505612879469113749589, 10.73252545284597420117471941249, 12.00492380768242560403405660627