| L(s) = 1 | + (−0.759 + 1.19i)2-s + (0.467 − 1.28i)3-s + (−0.845 − 1.81i)4-s + (−1.73 + 0.305i)5-s + (1.17 + 1.53i)6-s + (1.91 − 3.31i)7-s + (2.80 + 0.368i)8-s + (0.868 + 0.728i)9-s + (0.952 − 2.29i)10-s + (2.26 − 1.30i)11-s + (−2.72 + 0.238i)12-s + (−2.03 − 5.60i)13-s + (2.49 + 4.80i)14-s + (−0.417 + 2.36i)15-s + (−2.56 + 3.06i)16-s + (2.13 − 1.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.537 + 0.843i)2-s + (0.269 − 0.741i)3-s + (−0.422 − 0.906i)4-s + (−0.775 + 0.136i)5-s + (0.480 + 0.625i)6-s + (0.723 − 1.25i)7-s + (0.991 + 0.130i)8-s + (0.289 + 0.242i)9-s + (0.301 − 0.727i)10-s + (0.682 − 0.394i)11-s + (−0.785 + 0.0689i)12-s + (−0.565 − 1.55i)13-s + (0.668 + 1.28i)14-s + (−0.107 + 0.611i)15-s + (−0.642 + 0.766i)16-s + (0.516 − 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.849827 - 0.210110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.849827 - 0.210110i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.759 - 1.19i)T \) |
| 19 | \( 1 + (0.713 - 4.30i)T \) |
| good | 3 | \( 1 + (-0.467 + 1.28i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (1.73 - 0.305i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.91 + 3.31i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.26 + 1.30i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.03 + 5.60i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.13 + 1.78i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (1.10 - 6.27i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.78 - 3.31i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.41 + 4.18i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.50iT - 37T^{2} \) |
| 41 | \( 1 + (6.78 + 2.46i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.24 + 0.571i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.73 - 3.13i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.49 - 0.264i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-4.81 - 5.74i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-5.64 - 0.995i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.11 + 2.51i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.236 - 1.34i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-10.5 - 3.83i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (6.51 + 2.37i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.89 - 3.40i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.91 - 2.51i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (9.33 - 7.83i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−13.26386702828116572857889102205, −11.87815405590872770627387927967, −10.69898040630199632426087683592, −9.851564201287857006363694310239, −8.070126752162284899480418644625, −7.79007266405758684596350798624, −6.99676510191874599005624723432, −5.39924666333255851978221843004, −3.90620516262816085563526445784, −1.17073866482735548724984208705,
2.17131520085594590113177892912, 3.95474435816385596214584264634, 4.71778488717347693547236901277, 6.94601729802350627424588800371, 8.386249384740091690780068103348, 9.046774648620436591218059674752, 9.861922275524563736811114376591, 11.19757704361351403530653931423, 11.97345557338860234765529946048, 12.47483465533156807765916105780
