| L(s) = 1 | + (−2.96 + 0.963i)2-s + (4.62 − 3.35i)4-s + (−0.439 + 1.35i)5-s + (2.23 + 3.08i)7-s + (−3.13 + 4.32i)8-s − 4.43i·10-s + (−0.450 + 10.9i)11-s + (−20.6 + 6.69i)13-s + (−9.60 − 6.97i)14-s + (−1.91 + 5.90i)16-s + (−16.1 − 5.24i)17-s + (−11.4 + 15.8i)19-s + (2.50 + 7.72i)20-s + (−9.25 − 33.0i)22-s + 4.82·23-s + ⋯ |
| L(s) = 1 | + (−1.48 + 0.481i)2-s + (1.15 − 0.839i)4-s + (−0.0878 + 0.270i)5-s + (0.319 + 0.440i)7-s + (−0.392 + 0.540i)8-s − 0.443i·10-s + (−0.0409 + 0.999i)11-s + (−1.58 + 0.515i)13-s + (−0.685 − 0.498i)14-s + (−0.119 + 0.369i)16-s + (−0.950 − 0.308i)17-s + (−0.604 + 0.831i)19-s + (0.125 + 0.386i)20-s + (−0.420 − 1.50i)22-s + 0.209·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.135050 + 0.402616i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.135050 + 0.402616i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (0.450 - 10.9i)T \) |
| good | 2 | \( 1 + (2.96 - 0.963i)T + (3.23 - 2.35i)T^{2} \) |
| 5 | \( 1 + (0.439 - 1.35i)T + (-20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (-2.23 - 3.08i)T + (-15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (20.6 - 6.69i)T + (136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (16.1 + 5.24i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (11.4 - 15.8i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 - 4.82T + 529T^{2} \) |
| 29 | \( 1 + (-26.7 - 36.7i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (6.47 + 19.9i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-17.4 + 12.6i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (21.5 - 29.6i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 10.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (53.7 + 39.0i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-25.4 - 78.4i)T + (-2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-51.7 + 37.6i)T + (1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 3.37i)T + (3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 - 22.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + (13.7 - 42.2i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (41.1 + 56.6i)T + (-1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-132. + 43.1i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (106. + 34.6i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 - 63.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-31.2 - 96.1i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−14.65588998262985661327071298747, −12.84433437233598328886421119497, −11.70609554403350428460482902750, −10.47948077592792023394030879332, −9.595340864172459924185074541187, −8.655932612213196905045832291551, −7.44117392733738318662527057689, −6.68715602566551895080871152501, −4.78066245049703950730758969566, −2.10551867134037682844109217015,
0.50240853219551537343167751680, 2.54723052953527511893607371862, 4.79167240921289768414545688612, 6.82815548171415434114837975513, 8.058378266748296880879772968415, 8.816975622626588045360604904829, 10.02243082410166480259793035369, 10.84712941049301219676372453087, 11.77149809453650366844660018211, 13.01879118935965636178813746872
