| L(s) = 1 | + (−2.98 − 0.800i)2-s + (3.86 + 3.47i)3-s + (1.36 + 0.787i)4-s + (−6.51 + 6.51i)5-s + (−8.76 − 13.4i)6-s + (4.63 + 17.2i)7-s + (14.0 + 14.0i)8-s + (2.86 + 26.8i)9-s + (24.6 − 14.2i)10-s + (−10.6 + 39.7i)11-s + (2.53 + 7.77i)12-s + (−1.70 − 46.8i)13-s − 55.3i·14-s + (−47.8 + 2.54i)15-s + (−37.0 − 64.1i)16-s + (18.9 − 32.7i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 0.283i)2-s + (0.743 + 0.668i)3-s + (0.170 + 0.0984i)4-s + (−0.582 + 0.582i)5-s + (−0.596 − 0.917i)6-s + (0.250 + 0.933i)7-s + (0.621 + 0.621i)8-s + (0.106 + 0.994i)9-s + (0.780 − 0.450i)10-s + (−0.292 + 1.09i)11-s + (0.0609 + 0.187i)12-s + (−0.0363 − 0.999i)13-s − 1.05i·14-s + (−0.822 + 0.0438i)15-s + (−0.579 − 1.00i)16-s + (0.270 − 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.586842 + 0.513238i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586842 + 0.513238i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.86 - 3.47i)T \) |
| 13 | \( 1 + (1.70 + 46.8i)T \) |
| good | 2 | \( 1 + (2.98 + 0.800i)T + (6.92 + 4i)T^{2} \) |
| 5 | \( 1 + (6.51 - 6.51i)T - 125iT^{2} \) |
| 7 | \( 1 + (-4.63 - 17.2i)T + (-297. + 171.5i)T^{2} \) |
| 11 | \( 1 + (10.6 - 39.7i)T + (-1.15e3 - 665.5i)T^{2} \) |
| 17 | \( 1 + (-18.9 + 32.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-11.2 + 3.02i)T + (5.94e3 - 3.42e3i)T^{2} \) |
| 23 | \( 1 + (36.7 + 63.6i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-261. + 150. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-144. - 144. i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (53.9 + 14.4i)T + (4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (297. + 79.6i)T + (5.96e4 + 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-136. - 78.8i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-260. - 260. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 - 411. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-442. + 118. i)T + (1.77e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-36.3 + 62.9i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-231. + 862. i)T + (-2.60e5 - 1.50e5i)T^{2} \) |
| 71 | \( 1 + (180. + 672. i)T + (-3.09e5 + 1.78e5i)T^{2} \) |
| 73 | \( 1 + (366. - 366. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 764.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (332. - 332. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + (-242. + 904. i)T + (-6.10e5 - 3.52e5i)T^{2} \) |
| 97 | \( 1 + (912. - 244. i)T + (7.90e5 - 4.56e5i)T^{2} \) |
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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
−15.77351172164397373642810724305, −15.12644130346238346730555638880, −13.95969660441184930495398980870, −12.11450254867580307218844505492, −10.64518922517440186917101974035, −9.827185255159894304170606480409, −8.556941292501814400739902578172, −7.63297298953751425355345265728, −4.88686862145065586238157069963, −2.62626791535506574579121970365,
0.929088059882616547579084464031, 4.00048457888979763361800302610, 6.88114501516772872346404618828, 8.085382306108854968270402371134, 8.696331798957666391595521498173, 10.18405900129824312302667077166, 11.85769658686614534151525230362, 13.32294896600099778140241426901, 14.10968799239779015528334073953, 15.86203872514777987444526555761
