| L(s) = 1 | + (−1.99 − 0.126i)2-s + 2.39·3-s + (3.96 + 0.506i)4-s + (−2.73 + 1.98i)5-s + (−4.77 − 0.303i)6-s + (−2.96 + 9.12i)7-s + (−7.85 − 1.51i)8-s − 3.27·9-s + (5.71 − 3.61i)10-s + (2.71 + 1.97i)11-s + (9.49 + 1.21i)12-s + (−17.2 + 5.60i)13-s + (7.07 − 17.8i)14-s + (−6.54 + 4.75i)15-s + (15.4 + 4.02i)16-s + (12.0 − 16.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 − 0.0634i)2-s + 0.797·3-s + (0.991 + 0.126i)4-s + (−0.547 + 0.397i)5-s + (−0.795 − 0.0506i)6-s + (−0.423 + 1.30i)7-s + (−0.981 − 0.189i)8-s − 0.364·9-s + (0.571 − 0.361i)10-s + (0.246 + 0.179i)11-s + (0.791 + 0.100i)12-s + (−1.32 + 0.430i)13-s + (0.505 − 1.27i)14-s + (−0.436 + 0.316i)15-s + (0.967 + 0.251i)16-s + (0.706 − 0.972i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.373154 + 0.614058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.373154 + 0.614058i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 + 0.126i)T \) |
| 41 | \( 1 + (-37.4 + 16.7i)T \) |
| good | 3 | \( 1 - 2.39T + 9T^{2} \) |
| 5 | \( 1 + (2.73 - 1.98i)T + (7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (2.96 - 9.12i)T + (-39.6 - 28.8i)T^{2} \) |
| 11 | \( 1 + (-2.71 - 1.97i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (17.2 - 5.60i)T + (136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (-12.0 + 16.5i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (4.37 - 13.4i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-1.61 + 0.526i)T + (427. - 310. i)T^{2} \) |
| 29 | \( 1 + (-23.0 - 31.7i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (31.7 - 43.6i)T + (-296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-36.0 + 26.1i)T + (423. - 1.30e3i)T^{2} \) |
| 43 | \( 1 + (-61.7 + 20.0i)T + (1.49e3 - 1.08e3i)T^{2} \) |
| 47 | \( 1 + (-0.586 - 1.80i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (17.3 + 23.8i)T + (-868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (19.4 - 6.33i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-4.21 + 12.9i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (15.9 - 11.6i)T + (1.38e3 - 4.26e3i)T^{2} \) |
| 71 | \( 1 + (41.1 + 29.9i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + 30.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 145.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 74.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-112. - 36.6i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (34.9 + 48.1i)T + (-2.90e3 + 8.94e3i)T^{2} \) |
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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
−12.39696709010037121436966936776, −12.02422306651368116477011207370, −10.83378662941596614773434702961, −9.419738833370844129328297331389, −9.098107977956428694182680852452, −7.87568419129259445421942365359, −7.04081424231854241532710076412, −5.57365912515415470592320749464, −3.25965898200984395400615493535, −2.34269117939339454948725766110,
0.52601771658062350123930556296, 2.69404546112414407993543535175, 4.11436217673572716939768137387, 6.14746466569981836659361770716, 7.63256331651748725612033472787, 7.929352542820767086017254660406, 9.236878094341604969637944990560, 10.02034209437014803291814512546, 11.05717528360297177226821986089, 12.16793498420630233963826686644
