L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.70 + 2.95i)5-s + (−2.62 − 0.358i)7-s − 0.999·8-s + (1.70 + 2.95i)10-s + (0.5 + 0.866i)11-s + 1.82·13-s + (−1.62 + 2.09i)14-s + (−0.5 + 0.866i)16-s + (−3.82 − 6.63i)17-s + (1.70 − 2.95i)19-s + 3.41·20-s + 0.999·22-s + (1.12 − 1.94i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.763 + 1.32i)5-s + (−0.990 − 0.135i)7-s − 0.353·8-s + (0.539 + 0.935i)10-s + (0.150 + 0.261i)11-s + 0.507·13-s + (−0.433 + 0.558i)14-s + (−0.125 + 0.216i)16-s + (−0.928 − 1.60i)17-s + (0.391 − 0.678i)19-s + 0.763·20-s + 0.213·22-s + (0.233 − 0.404i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.153080900\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.153080900\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.62 + 0.358i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (1.70 - 2.95i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 + (3.82 + 6.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.70 + 2.95i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.12 + 1.94i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.65T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.29 + 5.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + (-3.24 + 5.61i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.94 + 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.20 + 7.28i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.08 - 5.34i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.62 + 9.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.07T + 71T^{2} \) |
| 73 | \( 1 + (-3.29 - 5.70i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.37 + 4.11i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + (2.24 - 3.88i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.516367604289941910145732344039, −8.781892845747340239423792698563, −7.48933789252474001631605144195, −6.81841294357863948575366180259, −6.31169845196690366901169820586, −4.91187433735736398387896429515, −4.01575462380281164272132198203, −3.04187915196937989360544371497, −2.59587017626661169940388430426, −0.49158167002211527605023199551,
1.13235652129282728469954378969, 3.02381880984752048594946359772, 4.09586248631227615082460066769, 4.52770823154717838700104514266, 5.90730961329501891166193816931, 6.20405512871942641918250070681, 7.43011093025484928385288641421, 8.285686379775834893414984967023, 8.731391326506155723710740219683, 9.495906644236889521013785242040