| L(s) = 1 | + (2.00 − 1.15i)3-s + (−67.5 − 116. i)5-s − 157.·7-s + (−361. + 626. i)9-s + 663.·11-s + (1.63e3 + 941. i)13-s + (−271. − 156. i)15-s + (938. + 1.62e3i)17-s + (−2.95e3 + 6.18e3i)19-s + (−316. + 182. i)21-s + (−4.91e3 + 8.50e3i)23-s + (−1.30e3 + 2.25e3i)25-s + 3.36e3i·27-s + (1.26e4 + 7.27e3i)29-s + 4.45e4i·31-s + ⋯ |
| L(s) = 1 | + (0.0743 − 0.0429i)3-s + (−0.540 − 0.935i)5-s − 0.459·7-s + (−0.496 + 0.859i)9-s + 0.498·11-s + (0.742 + 0.428i)13-s + (−0.0802 − 0.0463i)15-s + (0.191 + 0.330i)17-s + (−0.430 + 0.902i)19-s + (−0.0341 + 0.0197i)21-s + (−0.403 + 0.699i)23-s + (−0.0832 + 0.144i)25-s + 0.171i·27-s + (0.516 + 0.298i)29-s + 1.49i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0196 - 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0196 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.730003 + 0.715779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.730003 + 0.715779i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.95e3 - 6.18e3i)T \) |
| good | 3 | \( 1 + (-2.00 + 1.15i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (67.5 + 116. i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + 157.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 663.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-1.63e3 - 941. i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (-938. - 1.62e3i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 23 | \( 1 + (4.91e3 - 8.50e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.26e4 - 7.27e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 - 4.45e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 5.50e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (3.54e4 - 2.04e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (1.48e4 + 2.57e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.01e4 - 1.76e4i)T + (-5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.12e5 + 6.48e4i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-2.94e5 + 1.70e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (4.70e4 - 8.15e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.54e5 - 8.89e4i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-1.12e5 + 6.50e4i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (3.64e5 + 6.30e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.04e5 - 1.75e5i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 1.06e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.83e5 - 1.05e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-7.36e5 + 4.25e5i)T + (4.16e11 - 7.21e11i)T^{2} \) |
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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
−13.50468831257850794090740605952, −12.45758811242096764000639620850, −11.52202876917982283014551211390, −10.21489679911244700094632475934, −8.780584020181309014788416596515, −8.071734297170000426553282928600, −6.41189985192447552521783217129, −4.95534120914199771579333735027, −3.56168677023192385658424786636, −1.47346970664464864385399208950,
0.40063367897167111406575995821, 2.84394647311237911193417536220, 3.95025990730742456984677766770, 6.05343933884503800428806291949, 6.98940778608064202548355374749, 8.446082615090994952649702531162, 9.643213582609908651326641231108, 10.94768567217091776865870717249, 11.76790569780315283741155591988, 13.01900461335054832602143414742
