| L(s) = 1 | − 32·9-s + 128·17-s + 112·25-s + 160·41-s − 28·49-s − 368·73-s + 484·81-s + 112·89-s + 64·97-s − 512·113-s − 808·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4.09e3·153-s + 157-s + 163-s + 167-s + 400·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.55·9-s + 7.52·17-s + 4.47·25-s + 3.90·41-s − 4/7·49-s − 5.04·73-s + 5.97·81-s + 1.25·89-s + 0.659·97-s − 4.53·113-s − 6.67·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 26.7·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.36·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.718136793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718136793\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 + 16 T^{2} + 142 T^{4} + 16 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 5 | \( ( 1 - 56 T^{2} + 78 p^{2} T^{4} - 56 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 404 T^{2} + 68742 T^{4} + 404 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 200 T^{2} + 30078 T^{4} - 200 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 32 T + 750 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 464 T^{2} + 112782 T^{4} + 464 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1508 T^{2} + 1042182 T^{4} - 1508 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 964 T^{2} + 1109286 T^{4} + 964 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 716 T^{2} - 69018 T^{4} + 716 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 140 p T^{2} + 10434918 T^{4} - 140 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 40 T + 3006 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 1940 T^{2} + 7476102 T^{4} + 1940 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4084 T^{2} + 12949350 T^{4} - 4084 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 3502 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 - 2032 T^{2} + 19804878 T^{4} - 2032 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 8632 T^{2} + 40857438 T^{4} - 8632 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 572 T^{2} + 33416742 T^{4} - 572 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 13700 T^{2} + 92907462 T^{4} - 13700 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 92 T + 4374 T^{2} + 92 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 22084 T^{2} + 198084102 T^{4} - 22084 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 4016 T^{2} + 20075982 T^{4} + 4016 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 28 T + 10662 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 16 T + 18798 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 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
−3.68853799711768750247619364946, −3.65110889713663798474162157648, −3.24879952207539397373502357140, −3.08872236728563116851490687518, −3.07288229442833665530584593324, −3.06736299103360995540993887926, −3.01404507755819207920625259227, −2.90665778128595087716454746755, −2.73110011080149117459981149321, −2.71628842847567944272562848579, −2.71564443536374792075894073815, −2.34318547142870693426236689611, −2.12082344848415875746451895342, −2.10949106753760469333092225138, −1.88975905708310978038642419246, −1.41926080105504027289483416972, −1.27879964490658552314214732299, −1.21691182693337037923073451690, −1.21269457439119492850873743029, −1.05385587848405438384032999691, −0.969182016978440603759608698024, −0.926461724798474877060656890549, −0.48338797271428999068369673679, −0.31517257052242196927942104147, −0.080583288263052632131702314041,
0.080583288263052632131702314041, 0.31517257052242196927942104147, 0.48338797271428999068369673679, 0.926461724798474877060656890549, 0.969182016978440603759608698024, 1.05385587848405438384032999691, 1.21269457439119492850873743029, 1.21691182693337037923073451690, 1.27879964490658552314214732299, 1.41926080105504027289483416972, 1.88975905708310978038642419246, 2.10949106753760469333092225138, 2.12082344848415875746451895342, 2.34318547142870693426236689611, 2.71564443536374792075894073815, 2.71628842847567944272562848579, 2.73110011080149117459981149321, 2.90665778128595087716454746755, 3.01404507755819207920625259227, 3.06736299103360995540993887926, 3.07288229442833665530584593324, 3.08872236728563116851490687518, 3.24879952207539397373502357140, 3.65110889713663798474162157648, 3.68853799711768750247619364946
Plot not available for L-functions of degree greater than 10.
