| L(s) = 1 | + (1.69 − 2.32i)2-s + (−1.38 − 2.66i)3-s + (−1.32 − 4.07i)4-s + (−2.90 − 4.00i)5-s + (−8.53 − 1.27i)6-s + (−3.45 − 10.6i)7-s + (−0.767 − 0.249i)8-s + (−5.16 + 7.37i)9-s − 14.2·10-s + (−9.00 + 9.15i)12-s + (13.6 + 9.93i)13-s + (−30.5 − 9.94i)14-s + (−6.62 + 13.2i)15-s + (11.9 − 8.69i)16-s + (−0.0367 − 0.0506i)17-s + (8.42 + 24.4i)18-s + ⋯ |
| L(s) = 1 | + (0.845 − 1.16i)2-s + (−0.461 − 0.887i)3-s + (−0.330 − 1.01i)4-s + (−0.581 − 0.800i)5-s + (−1.42 − 0.212i)6-s + (−0.493 − 1.51i)7-s + (−0.0959 − 0.0311i)8-s + (−0.573 + 0.818i)9-s − 1.42·10-s + (−0.750 + 0.763i)12-s + (1.05 + 0.764i)13-s + (−2.18 − 0.710i)14-s + (−0.441 + 0.885i)15-s + (0.748 − 0.543i)16-s + (−0.00216 − 0.00297i)17-s + (0.467 + 1.36i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.791194 + 1.44910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.791194 + 1.44910i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.38 + 2.66i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.69 + 2.32i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (2.90 + 4.00i)T + (-7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (3.45 + 10.6i)T + (-39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (-13.6 - 9.93i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (0.0367 + 0.0506i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (-1.66 + 5.11i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 - 8.69iT - 529T^{2} \) |
| 29 | \( 1 + (47.5 - 15.4i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-19.8 - 14.4i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (12.1 + 37.3i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (41.7 + 13.5i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 - 0.201T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-14.1 - 4.60i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-57.5 + 79.2i)T + (-868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (13.7 - 4.47i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (23.7 - 17.2i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 - 82.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-15.6 - 21.6i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-4.31 - 13.2i)T + (-4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (40.6 + 29.4i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (35.4 + 48.7i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 + 40.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-24.9 - 18.1i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−11.04077667572701963233032546817, −10.17223442551311734442004964922, −8.752194224587110891873211096586, −7.59480691087268459392395793073, −6.77037162828158281037027194096, −5.37995401913914912949020337100, −4.26275361805668144644720904020, −3.51150422509201890298230501747, −1.68930705923999563010371736914, −0.62531982805033006290839180202,
3.07145327015844481611202749813, 3.93164829080665250879279029763, 5.28504914739116628480079285226, 5.92771055433824607002537160163, 6.62412427462723430890483969460, 7.941002670316678838216872662165, 8.861817313409928740326750255644, 10.03141662397853562689005673989, 11.00775543278075681728014306284, 11.84143455660684255842977459828
