L(s) = 1 | − 4.87·2-s + (−1.11 − 5.07i)3-s + 15.7·4-s + (3.10 − 5.37i)5-s + (5.43 + 24.7i)6-s + (17.8 − 4.93i)7-s − 37.6·8-s + (−24.5 + 11.3i)9-s + (−15.1 + 26.2i)10-s + (−27.7 − 48.0i)11-s + (−17.5 − 79.8i)12-s + (−6.74 − 11.6i)13-s + (−86.9 + 24.0i)14-s + (−30.7 − 9.75i)15-s + 57.5·16-s + (−43.1 + 74.7i)17-s + ⋯ |
L(s) = 1 | − 1.72·2-s + (−0.214 − 0.976i)3-s + 1.96·4-s + (0.277 − 0.481i)5-s + (0.369 + 1.68i)6-s + (0.963 − 0.266i)7-s − 1.66·8-s + (−0.907 + 0.419i)9-s + (−0.478 + 0.828i)10-s + (−0.761 − 1.31i)11-s + (−0.422 − 1.92i)12-s + (−0.143 − 0.249i)13-s + (−1.65 + 0.459i)14-s + (−0.529 − 0.167i)15-s + 0.899·16-s + (−0.616 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.107756 - 0.455133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.107756 - 0.455133i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.11 + 5.07i)T \) |
| 7 | \( 1 + (-17.8 + 4.93i)T \) |
good | 2 | \( 1 + 4.87T + 8T^{2} \) |
| 5 | \( 1 + (-3.10 + 5.37i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (27.7 + 48.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (6.74 + 11.6i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (43.1 - 74.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (42.1 + 72.9i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (5.26 - 9.12i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-76.2 + 132. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 254.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (172. + 298. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (57.9 + 100. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-46.7 + 80.9i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 319.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (136. - 236. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 - 350.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 291.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 833.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 771.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (140. - 242. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 - 345.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (62.9 - 108. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (163. + 282. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-124. + 215. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.85838267560293501088226757098, −12.72572134655455360378796445606, −11.14743435888883901990237083676, −10.77331550707515710948219984159, −8.855081652112060221501419655368, −8.257082856021267878227868813870, −7.17863573309778852327120667590, −5.64982539950900235465686599041, −2.05528013616733106929551646926, −0.55022030973544434946744582911,
2.25685318631728263826193904210, 4.93621818093769950374329636359, 6.85705219289981137872519128615, 8.175271913469319972304956694711, 9.299923404790464618348705768989, 10.25952212751347411145400865777, 10.94598392785664374870802541858, 12.08253132481511932400135855608, 14.37870574002172191867410879766, 15.28347424164635529360632475064