L(s) = 1 | + (−1.72 − 0.133i)3-s − 4.22·5-s + (−2.37 − 1.15i)7-s + (2.96 + 0.460i)9-s + 1.92·11-s + (−0.291 − 0.504i)13-s + (7.29 + 0.562i)15-s + (3.61 + 6.25i)17-s + (2.10 − 3.64i)19-s + (3.95 + 2.31i)21-s + 1.27·23-s + 12.8·25-s + (−5.05 − 1.18i)27-s + (−4.20 + 7.27i)29-s + (0.476 − 0.824i)31-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0769i)3-s − 1.88·5-s + (−0.898 − 0.438i)7-s + (0.988 + 0.153i)9-s + 0.581·11-s + (−0.0808 − 0.140i)13-s + (1.88 + 0.145i)15-s + (0.875 + 1.51i)17-s + (0.482 − 0.835i)19-s + (0.862 + 0.506i)21-s + 0.266·23-s + 2.56·25-s + (−0.973 − 0.228i)27-s + (−0.780 + 1.35i)29-s + (0.0855 − 0.148i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.563738 + 0.0933880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.563738 + 0.0933880i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 + 0.133i)T \) |
| 7 | \( 1 + (2.37 + 1.15i)T \) |
good | 5 | \( 1 + 4.22T + 5T^{2} \) |
| 11 | \( 1 - 1.92T + 11T^{2} \) |
| 13 | \( 1 + (0.291 + 0.504i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.61 - 6.25i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.10 + 3.64i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 + (4.20 - 7.27i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.476 + 0.824i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.03 + 5.25i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.31 - 2.27i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.442 + 0.766i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.88 + 4.99i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.962 + 1.66i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.27 + 3.94i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.29 - 9.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.43 + 4.21i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + (-0.446 - 0.772i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.93 - 10.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.24 - 9.08i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.87 + 6.71i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.98 - 3.44i)T + (-48.5 - 84.0i)T^{2} \) |
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
−11.08193052222446449395890791644, −10.38466328387851304810678129774, −9.233838407655682674814245353511, −8.033354262772195385256844468667, −7.21878569267140984092937428039, −6.57106674086578335875774279151, −5.28153311831539665158339296807, −4.05237893157568640823701596594, −3.49206197427613962359615573876, −0.834125618975644538502094742088,
0.61925304214287558081790021112, 3.21775095225591091225620571640, 4.09596258707585516849540996857, 5.15531750731836962183228786590, 6.32347069440460156903807016932, 7.24576134409447971781268276789, 7.913213547258601945339693870054, 9.264181684218671813758245851408, 9.993835846361120890222544113164, 11.24752251620071133239241736320