| L(s) = 1 | + (2.50 + 6.87i)5-s + (−1.62 − 9.22i)7-s + (4.02 − 11.0i)11-s + (17.4 + 14.6i)13-s + (13.5 + 7.81i)17-s + (9.08 + 15.7i)19-s + (−23.0 − 4.06i)23-s + (−21.8 + 18.3i)25-s + (24.8 + 29.6i)29-s + (−4.47 + 25.3i)31-s + (59.3 − 34.2i)35-s + (1.45 − 2.52i)37-s + (26.1 − 31.1i)41-s + (35.7 + 12.9i)43-s + (18.5 − 3.26i)47-s + ⋯ |
| L(s) = 1 | + (0.500 + 1.37i)5-s + (−0.232 − 1.31i)7-s + (0.366 − 1.00i)11-s + (1.34 + 1.12i)13-s + (0.795 + 0.459i)17-s + (0.478 + 0.828i)19-s + (−1.00 − 0.176i)23-s + (−0.872 + 0.732i)25-s + (0.858 + 1.02i)29-s + (−0.144 + 0.818i)31-s + (1.69 − 0.979i)35-s + (0.0394 − 0.0682i)37-s + (0.637 − 0.760i)41-s + (0.830 + 0.302i)43-s + (0.394 − 0.0694i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.81458 + 0.499814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81458 + 0.499814i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.50 - 6.87i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (1.62 + 9.22i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (-4.02 + 11.0i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-17.4 - 14.6i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-13.5 - 7.81i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-9.08 - 15.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (23.0 + 4.06i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-24.8 - 29.6i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (4.47 - 25.3i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-1.45 + 2.52i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-26.1 + 31.1i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-35.7 - 12.9i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-18.5 + 3.26i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 12.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (4.38 + 12.0i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (15.6 + 88.8i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (6.68 + 5.61i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (81.5 + 47.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (25.3 + 43.8i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (78.4 - 65.8i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-12.1 - 14.4i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-52.8 + 30.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (139. + 50.8i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.10753587567361606778330828747, −10.65052744709868715986663572582, −9.889102950121643568240219499095, −8.647905664917999355414348324688, −7.46779429938078205903857839704, −6.53478841235576590326525590153, −5.94854669854469236503866336971, −3.94757115251876695284017815783, −3.28809636343698644045216455228, −1.39424585647590113806548898582,
1.10865961960577063050696271150, 2.62536034915698155090844366153, 4.32927694024361380791525377902, 5.52288655624573187736477056377, 6.01065370496665955017510181143, 7.71974769101018133166018791972, 8.673405672034160203766666411026, 9.341010177403195574339895764986, 10.10801892698301967171657781754, 11.63718063229307670714316367113
