| L(s) = 1 | + (2.08 + 1.00i)2-s + (1.51 + 0.847i)3-s + (2.09 + 2.63i)4-s + (−1.46 + 1.36i)5-s + (2.29 + 3.28i)6-s + (−2.55 − 0.675i)7-s + (0.704 + 3.08i)8-s + (1.56 + 2.56i)9-s + (−4.42 + 1.36i)10-s + (2.74 − 1.86i)11-s + (0.938 + 5.75i)12-s + (3.06 − 2.08i)13-s + (−4.65 − 3.98i)14-s + (−3.36 + 0.811i)15-s + (−0.133 + 0.583i)16-s + (−5.17 − 0.779i)17-s + ⋯ |
| L(s) = 1 | + (1.47 + 0.710i)2-s + (0.871 + 0.489i)3-s + (1.04 + 1.31i)4-s + (−0.655 + 0.608i)5-s + (0.938 + 1.34i)6-s + (−0.966 − 0.255i)7-s + (0.248 + 1.09i)8-s + (0.520 + 0.853i)9-s + (−1.40 + 0.431i)10-s + (0.826 − 0.563i)11-s + (0.270 + 1.66i)12-s + (0.850 − 0.579i)13-s + (−1.24 − 1.06i)14-s + (−0.869 + 0.209i)15-s + (−0.0333 + 0.145i)16-s + (−1.25 − 0.189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0453 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0453 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.35353 + 2.46280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.35353 + 2.46280i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.51 - 0.847i)T \) |
| 7 | \( 1 + (2.55 + 0.675i)T \) |
| good | 2 | \( 1 + (-2.08 - 1.00i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (1.46 - 1.36i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (-2.74 + 1.86i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-3.06 + 2.08i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (5.17 + 0.779i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.930 + 1.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.869 - 2.21i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-4.97 - 0.749i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 + (3.33 + 8.50i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.0854 - 0.0263i)T + (33.8 + 23.0i)T^{2} \) |
| 43 | \( 1 + (-4.12 + 1.27i)T + (35.5 - 24.2i)T^{2} \) |
| 47 | \( 1 + (8.72 + 4.20i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-0.956 + 2.43i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (2.95 - 12.9i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-4.18 + 5.24i)T + (-13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 + 7.10T + 67T^{2} \) |
| 71 | \( 1 + (-6.33 - 7.94i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (6.80 + 4.64i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + (1.31 + 0.895i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (1.11 - 14.9i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (0.0150 + 0.0261i)T + (-48.5 + 84.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.44172581588031713447110558461, −10.67011052652207991012497052591, −9.406323581565630429881921207711, −8.473698155531142907212774614411, −7.27888445496921609795408425808, −6.71521558365250812439285313083, −5.63325536032287492294072956902, −4.22627243703783712210295710472, −3.61607494985392626504189907999, −2.90125740734474973057717062998,
1.63425536748124165920309194969, 2.93160667698366576935586807702, 3.94741858374569295177169430068, 4.52458650476013436119601259442, 6.25825071214665222172101592052, 6.73308291395731041540924429649, 8.306043795529290369279715869178, 9.034097148797855445735991666701, 10.08622078469667837616219066498, 11.40185304043742576087230838175
