L(s) = 1 | + (−44.4 − 8.44i)2-s − 243·3-s + (1.90e3 + 751. i)4-s + 1.13e4i·5-s + (1.08e4 + 2.05e3i)6-s + (−4.12e4 − 1.66e4i)7-s + (−7.83e4 − 4.94e4i)8-s + 5.90e4·9-s + (9.59e4 − 5.04e5i)10-s − 3.97e5i·11-s + (−4.62e5 − 1.82e5i)12-s + 2.06e6i·13-s + (1.69e6 + 1.08e6i)14-s − 2.75e6i·15-s + (3.06e6 + 2.86e6i)16-s + 7.16e6i·17-s + ⋯ |
L(s) = 1 | + (−0.982 − 0.186i)2-s − 0.577·3-s + (0.930 + 0.366i)4-s + 1.62i·5-s + (0.567 + 0.107i)6-s + (−0.927 − 0.373i)7-s + (−0.845 − 0.533i)8-s + 0.333·9-s + (0.303 − 1.59i)10-s − 0.743i·11-s + (−0.537 − 0.211i)12-s + 1.54i·13-s + (0.841 + 0.540i)14-s − 0.938i·15-s + (0.730 + 0.682i)16-s + 1.22i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.687i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.322068 + 0.808162i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.322068 + 0.808162i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (44.4 + 8.44i)T \) |
| 3 | \( 1 + 243T \) |
| 7 | \( 1 + (4.12e4 + 1.66e4i)T \) |
good | 5 | \( 1 - 1.13e4iT - 4.88e7T^{2} \) |
| 11 | \( 1 + 3.97e5iT - 2.85e11T^{2} \) |
| 13 | \( 1 - 2.06e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 - 7.16e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 1.91e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 7.66e6iT - 9.52e14T^{2} \) |
| 29 | \( 1 - 1.11e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.91e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.33e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.59e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 - 1.09e9iT - 9.29e17T^{2} \) |
| 47 | \( 1 + 2.07e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.51e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 1.86e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.15e9iT - 4.35e19T^{2} \) |
| 67 | \( 1 - 4.98e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 - 1.27e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 1.29e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 1.46e10iT - 7.47e20T^{2} \) |
| 83 | \( 1 - 2.39e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.95e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 6.05e10iT - 7.15e21T^{2} \) |
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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
−11.85245660450223864525146148833, −11.21338996390777169823234156337, −10.24024250843086165192786051802, −9.515645306938526710510339572693, −7.81882076409778498928001937446, −6.62669410837604589582408115604, −6.29199893110710238860667240503, −3.69522510631461493861948675467, −2.70956883893245933527010076838, −1.08838911359447198392463238826,
0.46879611525079984521097139343, 0.981087664367771063147197817003, 2.80697769671537071539967214962, 4.97038998482417264288676366920, 5.77623993608713046712273873846, 7.23819054892934055492455696570, 8.368713008192677186957180487972, 9.536062779037439686043365125183, 10.04949519265045179659876810473, 11.79667378819190193501703844619