L(s) = 1 | + (0.750 − 1.85i)2-s + (−2.15 − 2.08i)4-s + (0.926 − 0.375i)7-s + (−3.64 + 1.61i)8-s + (−0.518 − 0.855i)9-s + (−0.856 − 0.138i)11-s − 1.99i·14-s + (0.159 + 4.97i)16-s + (−1.97 + 0.319i)18-s + (−0.899 + 1.48i)22-s + (1.03 − 1.47i)23-s + (−0.801 + 0.598i)25-s + (−2.77 − 1.12i)28-s + (−0.161 − 0.545i)29-s + (5.59 + 2.05i)32-s + ⋯ |
L(s) = 1 | + (0.750 − 1.85i)2-s + (−2.15 − 2.08i)4-s + (0.926 − 0.375i)7-s + (−3.64 + 1.61i)8-s + (−0.518 − 0.855i)9-s + (−0.856 − 0.138i)11-s − 1.99i·14-s + (0.159 + 4.97i)16-s + (−1.97 + 0.319i)18-s + (−0.899 + 1.48i)22-s + (1.03 − 1.47i)23-s + (−0.801 + 0.598i)25-s + (−2.77 − 1.12i)28-s + (−0.161 − 0.545i)29-s + (5.59 + 2.05i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.199093741\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.199093741\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.926 + 0.375i)T \) |
| 197 | \( 1 + (-0.404 - 0.914i)T \) |
good | 2 | \( 1 + (-0.750 + 1.85i)T + (-0.718 - 0.695i)T^{2} \) |
| 3 | \( 1 + (0.518 + 0.855i)T^{2} \) |
| 5 | \( 1 + (0.801 - 0.598i)T^{2} \) |
| 11 | \( 1 + (0.856 + 0.138i)T + (0.949 + 0.315i)T^{2} \) |
| 13 | \( 1 + (0.926 + 0.375i)T^{2} \) |
| 17 | \( 1 + (-0.981 + 0.191i)T^{2} \) |
| 19 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 23 | \( 1 + (-1.03 + 1.47i)T + (-0.345 - 0.938i)T^{2} \) |
| 29 | \( 1 + (0.161 + 0.545i)T + (-0.838 + 0.545i)T^{2} \) |
| 31 | \( 1 + (-0.0960 + 0.995i)T^{2} \) |
| 37 | \( 1 + (0.00615 - 0.191i)T + (-0.997 - 0.0640i)T^{2} \) |
| 41 | \( 1 + (0.981 - 0.191i)T^{2} \) |
| 43 | \( 1 + (0.255 - 1.58i)T + (-0.949 - 0.315i)T^{2} \) |
| 47 | \( 1 + (-0.871 - 0.490i)T^{2} \) |
| 53 | \( 1 + (-1.62 + 1.05i)T + (0.404 - 0.914i)T^{2} \) |
| 59 | \( 1 + (-0.991 + 0.127i)T^{2} \) |
| 61 | \( 1 + (-0.518 + 0.855i)T^{2} \) |
| 67 | \( 1 + (-0.328 + 1.25i)T + (-0.871 - 0.490i)T^{2} \) |
| 71 | \( 1 + (-1.18 + 0.721i)T + (0.462 - 0.886i)T^{2} \) |
| 73 | \( 1 + (-0.997 - 0.0640i)T^{2} \) |
| 79 | \( 1 + (0.343 - 1.03i)T + (-0.801 - 0.598i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.0960 - 0.995i)T^{2} \) |
| 97 | \( 1 + (-0.967 - 0.253i)T^{2} \) |
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
−9.610076502038039335679350074719, −8.770371432982637504745721965518, −8.069421901231444641365510550803, −6.47623957899907906282797791743, −5.48427007079162774898219710963, −4.82701718471694214313314543826, −3.95445071842792541335204493524, −3.03882621547790655156221804416, −2.13366187371401782506162392128, −0.806261202310483711483787913356,
2.55055266519041570710722758300, 3.77203213125142803409594716767, 4.85362382497774369921588358577, 5.36230538898508920879955443343, 5.88243319413584629821340435356, 7.27833587064411329780677339528, 7.52681939010585063221663995773, 8.461805158265181647922867265534, 8.837484994580412520965935396910, 10.00339335923806983784425657342