L(s) = 1 | + 38.1i·3-s + 102.·5-s + 512.·7-s − 726.·9-s + 2.41e3·11-s − 3.77e3i·13-s + 3.90e3i·15-s + 1.56e3·17-s + (−2.73e3 + 6.28e3i)19-s + 1.95e4i·21-s − 626.·23-s − 5.12e3·25-s + 96.4i·27-s + 1.57e4i·29-s − 2.61e4i·31-s + ⋯ |
L(s) = 1 | + 1.41i·3-s + 0.819·5-s + 1.49·7-s − 0.996·9-s + 1.81·11-s − 1.71i·13-s + 1.15i·15-s + 0.318·17-s + (−0.398 + 0.916i)19-s + 2.11i·21-s − 0.0514·23-s − 0.327·25-s + 0.00489i·27-s + 0.643i·29-s − 0.879i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.916i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.398 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.26934 + 1.48750i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.26934 + 1.48750i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.73e3 - 6.28e3i)T \) |
good | 3 | \( 1 - 38.1iT - 729T^{2} \) |
| 5 | \( 1 - 102.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 512.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 2.41e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.77e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 1.56e3T + 2.41e7T^{2} \) |
| 23 | \( 1 + 626.T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.57e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.61e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 4.94e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.19e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.33e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.86e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.04e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.82e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.53e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 5.19e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.68e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.85e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.73e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 1.27e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 6.07e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 2.82e5iT - 8.32e11T^{2} \) |
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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
−13.90326215620337080595992074055, −12.19318625533498406139055447936, −11.00241644448962703920474202194, −10.16560652443244514593526788983, −9.178763416242989429124200326912, −8.008859832895877256123297594320, −5.94072944848448117514791268820, −4.87917508834811940726653087756, −3.63653121635304310063833698014, −1.52203592176866892054030876066,
1.38481190006585617389101392727, 1.90481307981278954736588639204, 4.49790604896557482901876417388, 6.25119215300870954724100826257, 7.03246730815842791717160387625, 8.407286572326488424995337547529, 9.453854943780647633886822945444, 11.45034277358976109611665016865, 11.80317066852144588193323930476, 13.23423979998528941774082824942