L(s) = 1 | + (173. − 299. i)5-s + (334. − 843. i)7-s + (437. + 758. i)11-s − 507.·13-s + (−9.80e3 − 1.69e4i)17-s + (1.77e4 − 3.07e4i)19-s + (4.87e4 − 8.44e4i)23-s + (−2.08e4 − 3.61e4i)25-s + 5.66e4·29-s + (2.63e4 + 4.56e4i)31-s + (−1.95e5 − 2.46e5i)35-s + (−1.71e5 + 2.97e5i)37-s + 2.79e5·41-s + 7.37e4·43-s + (−5.14e5 + 8.91e5i)47-s + ⋯ |
L(s) = 1 | + (0.619 − 1.07i)5-s + (0.368 − 0.929i)7-s + (0.0991 + 0.171i)11-s − 0.0640·13-s + (−0.483 − 0.838i)17-s + (0.594 − 1.02i)19-s + (0.835 − 1.44i)23-s + (−0.267 − 0.463i)25-s + 0.431·29-s + (0.158 + 0.274i)31-s + (−0.769 − 0.970i)35-s + (−0.558 + 0.966i)37-s + 0.633·41-s + 0.141·43-s + (−0.723 + 1.25i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.231226625\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.231226625\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-334. + 843. i)T \) |
good | 5 | \( 1 + (-173. + 299. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-437. - 758. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 507.T + 6.27e7T^{2} \) |
| 17 | \( 1 + (9.80e3 + 1.69e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.77e4 + 3.07e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-4.87e4 + 8.44e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 5.66e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-2.63e4 - 4.56e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.71e5 - 2.97e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 2.79e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.37e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (5.14e5 - 8.91e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-1.86e5 - 3.23e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-5.20e5 - 9.01e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.61e5 + 2.79e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.50e6 + 2.60e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 2.97e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (1.26e6 + 2.19e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-9.36e5 + 1.62e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 3.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-5.03e6 + 8.71e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.52e7T + 8.07e13T^{2} \) |
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
−10.39955191634396893385697455423, −9.359020871729807959618980030306, −8.666900107294232573740181450290, −7.44261878206090896363331122839, −6.48119247477627307443423230937, −4.95943914345499152847084102581, −4.57238798708664737293221641690, −2.82645961850754058766981700778, −1.35730523348083663826886043989, −0.52240100222114391536864137158,
1.53828735351925056179386566235, 2.54934201963605474317865916551, 3.67153744586497430450868128493, 5.33908655695292302595980706844, 6.08125634015412382926745045311, 7.13620293644361412459075660088, 8.280606025757291142378929411087, 9.319501719917063353880873733598, 10.23967666644681893810278131487, 11.13537674916790671930301297016