| L(s) = 1 | + (2.24 − 1.62i)2-s + (−1.73 + 0.773i)3-s + (1.75 − 5.38i)4-s + (−0.5 − 0.866i)5-s + (−2.63 + 4.56i)6-s + (1.17 + 1.30i)7-s + (−3.13 − 9.65i)8-s + (0.415 − 0.461i)9-s + (−2.52 − 1.12i)10-s + (3.42 − 0.728i)11-s + (1.12 + 10.7i)12-s + (−0.611 + 5.82i)13-s + (4.76 + 1.01i)14-s + (1.53 + 1.11i)15-s + (−13.5 − 9.86i)16-s + (−1.08 − 0.230i)17-s + ⋯ |
| L(s) = 1 | + (1.58 − 1.15i)2-s + (−1.00 + 0.446i)3-s + (0.875 − 2.69i)4-s + (−0.223 − 0.387i)5-s + (−1.07 + 1.86i)6-s + (0.444 + 0.494i)7-s + (−1.10 − 3.41i)8-s + (0.138 − 0.153i)9-s + (−0.799 − 0.356i)10-s + (1.03 − 0.219i)11-s + (0.325 + 3.09i)12-s + (−0.169 + 1.61i)13-s + (1.27 + 0.270i)14-s + (0.397 + 0.288i)15-s + (−3.39 − 2.46i)16-s + (−0.263 − 0.0559i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0159 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0159 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30762 - 1.32860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30762 - 1.32860i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (1.91 + 5.22i)T \) |
| good | 2 | \( 1 + (-2.24 + 1.62i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (1.73 - 0.773i)T + (2.00 - 2.22i)T^{2} \) |
| 7 | \( 1 + (-1.17 - 1.30i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-3.42 + 0.728i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.611 - 5.82i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (1.08 + 0.230i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.381 - 3.63i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.769 - 2.36i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.505 + 0.367i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (2.35 - 4.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.89 + 3.51i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.283 + 2.69i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-1.74 - 1.26i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (4.45 - 4.94i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-8.48 + 3.77i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 3.24T + 61T^{2} \) |
| 67 | \( 1 + (-0.854 - 1.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.39 + 7.10i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (11.3 - 2.40i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (9.04 + 1.92i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-5.44 - 2.42i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-0.455 + 1.40i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.0386 - 0.119i)T + (-78.4 - 57.0i)T^{2} \) |
| 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
−12.29253813903836998297003238088, −11.62896380740111679995487124922, −11.40933566813644570392067281716, −10.15603162023916025770183367320, −9.083017546010235159272718141452, −6.63207148363564205373330392059, −5.65843322108840364555055608805, −4.72072052058183722464562450970, −3.85562072640161041515524015210, −1.80490286996569391639125459150,
3.25634445287518612697171096238, 4.65232471470116571140851895609, 5.63299904280902576370255509431, 6.68395019262297721516539898857, 7.27471446704161612048286193138, 8.490006685722980103236966848263, 10.81339109047624277203622200406, 11.63004567183055497376851956956, 12.44234790605639418374065592601, 13.17946391363781874705435202879
