| L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−1.69 − 1.96i)5-s + (−1.04 − 0.668i)7-s + (−0.415 + 0.909i)8-s + (−2.18 + 1.40i)10-s + (−0.0886 − 0.616i)11-s + (−1.11 + 0.716i)13-s + (−0.809 + 0.934i)14-s + (0.841 + 0.540i)16-s + (−4.51 + 1.32i)17-s + (−3.63 − 1.06i)19-s + (1.07 + 2.35i)20-s − 0.622·22-s + (−3.35 + 3.42i)23-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.479 − 0.140i)4-s + (−0.759 − 0.876i)5-s + (−0.393 − 0.252i)7-s + (−0.146 + 0.321i)8-s + (−0.690 + 0.443i)10-s + (−0.0267 − 0.185i)11-s + (−0.309 + 0.198i)13-s + (−0.216 + 0.249i)14-s + (0.210 + 0.135i)16-s + (−1.09 + 0.321i)17-s + (−0.834 − 0.244i)19-s + (0.240 + 0.527i)20-s − 0.132·22-s + (−0.698 + 0.715i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0828359 + 0.532355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0828359 + 0.532355i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (3.35 - 3.42i)T \) |
| good | 5 | \( 1 + (1.69 + 1.96i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (1.04 + 0.668i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.0886 + 0.616i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.11 - 0.716i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.51 - 1.32i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (3.63 + 1.06i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (1.06 - 0.313i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.92 + 8.60i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 1.22i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (7.75 + 8.95i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.697 + 1.52i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 + (-9.46 - 6.08i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-8.19 + 5.26i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (5.97 - 13.0i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.871 + 6.05i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.923 + 6.42i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-12.5 - 3.68i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (10.1 - 6.53i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.83 + 5.58i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (1.40 + 3.07i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (7.49 + 8.65i)T + (-13.8 + 96.0i)T^{2} \) |
| 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
−10.83531903320243056026289232683, −9.885794323388599694148801954734, −8.899864337248714627036630692672, −8.239820082943529696633710454100, −7.03861546903089131193703285608, −5.74007860554082105500618966965, −4.44596962281584415395517678748, −3.85183387366883478043898944148, −2.19980737329544407475929884080, −0.32267721118445261242085146953,
2.65265086074556901285219879451, 3.88033910029252426744133833360, 4.95833980556395287543641379343, 6.41047190238350591146128002286, 6.86831305934342202461578776594, 7.962881735743695605409012661168, 8.745580508377212420403690718670, 9.927473323257868571693956884237, 10.76199913826157906418431298814, 11.76174701942550301602793798189
