| L(s) = 1 | + (1.89 + 0.625i)2-s + (3.21 + 2.37i)4-s + (−1.04 + 5.95i)5-s + (1.18 + 1.41i)7-s + (4.62 + 6.52i)8-s + (−5.71 + 10.6i)10-s + (−14.9 + 2.63i)11-s + (−6.81 + 2.48i)13-s + (1.36 + 3.42i)14-s + (4.69 + 15.2i)16-s + (−2.22 + 3.85i)17-s + (30.1 − 17.4i)19-s + (−17.5 + 16.6i)20-s + (−30.0 − 4.35i)22-s + (8.19 − 9.77i)23-s + ⋯ |
| L(s) = 1 | + (0.949 + 0.312i)2-s + (0.804 + 0.594i)4-s + (−0.209 + 1.19i)5-s + (0.169 + 0.201i)7-s + (0.577 + 0.816i)8-s + (−0.571 + 1.06i)10-s + (−1.36 + 0.239i)11-s + (−0.524 + 0.190i)13-s + (0.0977 + 0.244i)14-s + (0.293 + 0.955i)16-s + (−0.130 + 0.226i)17-s + (1.58 − 0.917i)19-s + (−0.876 + 0.832i)20-s + (−1.36 − 0.197i)22-s + (0.356 − 0.424i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.51968 + 2.12453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51968 + 2.12453i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.89 - 0.625i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.04 - 5.95i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-1.18 - 1.41i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (14.9 - 2.63i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (6.81 - 2.48i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (2.22 - 3.85i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-30.1 + 17.4i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-8.19 + 9.77i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-26.3 - 9.59i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (34.0 - 40.5i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-1.20 + 2.08i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-44.2 + 16.1i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (15.3 - 2.71i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-13.3 - 15.8i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 77.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-69.2 - 12.2i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-25.1 + 21.0i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-25.3 - 69.6i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (81.7 + 47.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (42.1 + 72.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (0.961 - 2.64i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-19.3 + 53.0i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (55.7 + 96.5i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (13.0 + 73.9i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.71841719602661505495045340916, −10.90026670572052763835721832851, −10.19205014590364025047317404323, −8.617740941257081195743140289205, −7.26584251297346020506495880267, −7.11227400723815257035842941098, −5.61900513780417368210274630377, −4.77981367878622523822520245569, −3.23429213402622998090703212507, −2.47710230207068025741403172501,
0.929549362646229764864617617258, 2.62664919145168290643443831914, 4.02271862530769974953174499470, 5.15133033914111974954527692643, 5.62190807289915856417563658356, 7.33854646622244326418409504839, 8.048975957643058243463276245591, 9.458806397310853884472666325014, 10.30878080376604582904129647863, 11.38960371493820601581974843226
