| L(s) = 1 | + (−2.05 − 2.83i)2-s + (−2.95 − 0.498i)3-s + (−2.55 + 7.85i)4-s + (−4.64 + 6.39i)5-s + (4.67 + 9.40i)6-s + (0.560 − 1.72i)7-s + (14.1 − 4.60i)8-s + (8.50 + 2.94i)9-s + 27.6·10-s + (11.4 − 21.9i)12-s + (−0.0573 + 0.0416i)13-s + (−6.04 + 1.96i)14-s + (16.9 − 16.5i)15-s + (−15.4 − 11.2i)16-s + (−2.15 + 2.96i)17-s + (−9.15 − 30.1i)18-s + ⋯ |
| L(s) = 1 | + (−1.02 − 1.41i)2-s + (−0.986 − 0.166i)3-s + (−0.637 + 1.96i)4-s + (−0.928 + 1.27i)5-s + (0.779 + 1.56i)6-s + (0.0801 − 0.246i)7-s + (1.77 − 0.575i)8-s + (0.944 + 0.327i)9-s + 2.76·10-s + (0.954 − 1.82i)12-s + (−0.00441 + 0.00320i)13-s + (−0.431 + 0.140i)14-s + (1.12 − 1.10i)15-s + (−0.967 − 0.703i)16-s + (−0.126 + 0.174i)17-s + (−0.508 − 1.67i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00929353 + 0.0715544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00929353 + 0.0715544i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.95 + 0.498i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (2.05 + 2.83i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (4.64 - 6.39i)T + (-7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (-0.560 + 1.72i)T + (-39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (0.0573 - 0.0416i)T + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (2.15 - 2.96i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (-8.13 - 25.0i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 - 6.84iT - 529T^{2} \) |
| 29 | \( 1 + (28.7 + 9.33i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (34.1 - 24.8i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-1.14 + 3.51i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (-67.5 + 21.9i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + 13.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-26.0 + 8.45i)T + (1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (12.7 + 17.5i)T + (-868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (75.8 + 24.6i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (64.0 + 46.5i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + 101.T + 4.48e3T^{2} \) |
| 71 | \( 1 + (27.2 - 37.5i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-35.6 + 109. i)T + (-4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (50.8 - 36.9i)T + (1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-49.6 + 68.3i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 45.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-77.8 + 56.5i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−10.78445669423528076368250263073, −10.28903380874536053655559680254, −9.210991784001351531702892956857, −7.71786588410889119753548340229, −7.40200591401687933521736694315, −5.97089482481472047502310665499, −4.13386538956303862951110506247, −3.27889289542370224598812299321, −1.72183394528121063937335387785, −0.07002057728810417005654112282,
0.985737933065735267391199209117, 4.31936048969098019961536932809, 5.16045419976479036463280670297, 5.97238477151657300367288990729, 7.21728270084062562328112926718, 7.78251731561191825609044277182, 9.078810331933217544978706929920, 9.257510533111916760201936596491, 10.69389212644029586336458723440, 11.55767832208959603044251717275
