| L(s) = 1 | − 4·5-s + 12·9-s + 24·25-s + 184·29-s − 256·41-s − 48·45-s − 208·49-s − 304·61-s + 90·81-s + 560·89-s − 296·101-s + 608·109-s + 272·121-s − 204·125-s + 127-s + 131-s + 137-s + 139-s − 736·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.18e3·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 4/5·5-s + 4/3·9-s + 0.959·25-s + 6.34·29-s − 6.24·41-s − 1.06·45-s − 4.24·49-s − 4.98·61-s + 10/9·81-s + 6.29·89-s − 2.93·101-s + 5.57·109-s + 2.24·121-s − 1.63·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 5.07·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 7.00·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4226111274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4226111274\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) |
| 5 | \( ( 1 + 2 T - 6 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + 104 T^{2} + 5454 T^{4} + 104 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 136 T^{2} + 28206 T^{4} - 136 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 592 T^{2} + 144510 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 112 T^{2} + 118878 T^{4} - 112 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 436 T^{2} + 275334 T^{4} - 436 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 76 p T^{2} + 1290726 T^{4} + 76 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 46 T + 1698 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1540 T^{2} + 1506054 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4720 T^{2} + 9299454 T^{4} - 4720 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 64 T + 2334 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 4868 T^{2} + 12528486 T^{4} + 4868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 5540 T^{2} + 15331014 T^{4} + 5540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 10480 T^{2} + 43220094 T^{4} - 10480 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 2744 T^{2} + 25695534 T^{4} + 2744 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 38 T + p^{2} T^{2} )^{8} \) |
| 67 | \( ( 1 + 8564 T^{2} + 44158854 T^{4} + 8564 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 3028 T^{2} - 19410330 T^{4} - 3028 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 17140 T^{2} + 129420582 T^{4} - 17140 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 22660 T^{2} + 205335174 T^{4} - 22660 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 23492 T^{2} + 230783910 T^{4} + 23492 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 140 T + 19830 T^{2} - 140 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 8252 T^{2} + 163205766 T^{4} + 8252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.04061733237711833512104368735, −4.00085748818378785444668807247, −3.99263539833658227662398411719, −3.50128552138462817245342063226, −3.42380788583507495960345614882, −3.35117014889612471114989863383, −3.19203358740664302770610764924, −3.09347233109709529023541219935, −3.07373298256087246620365470848, −2.96625263476086775497748392040, −2.96198889266008129213025067785, −2.71739902215957690524637652775, −2.30548607902819480527521022445, −2.10787977866707114990332604731, −1.91940676045167751169790719887, −1.91857364169984942664522626921, −1.69725631328386866928097046140, −1.68654242179898356822947790841, −1.41769419244204653472630830900, −1.15757574447309014126204355160, −0.991452294093540837484813135699, −0.75222436773444034239930372398, −0.69440435360764190619165303296, −0.38814877374767682492841676367, −0.04857835803796854480342185229,
0.04857835803796854480342185229, 0.38814877374767682492841676367, 0.69440435360764190619165303296, 0.75222436773444034239930372398, 0.991452294093540837484813135699, 1.15757574447309014126204355160, 1.41769419244204653472630830900, 1.68654242179898356822947790841, 1.69725631328386866928097046140, 1.91857364169984942664522626921, 1.91940676045167751169790719887, 2.10787977866707114990332604731, 2.30548607902819480527521022445, 2.71739902215957690524637652775, 2.96198889266008129213025067785, 2.96625263476086775497748392040, 3.07373298256087246620365470848, 3.09347233109709529023541219935, 3.19203358740664302770610764924, 3.35117014889612471114989863383, 3.42380788583507495960345614882, 3.50128552138462817245342063226, 3.99263539833658227662398411719, 4.00085748818378785444668807247, 4.04061733237711833512104368735
Plot not available for L-functions of degree greater than 10.
