| L(s) = 1 | + 2·4-s − 8·13-s − 16·37-s + 156·49-s − 16·52-s + 40·61-s + 24·64-s − 240·73-s − 120·97-s + 56·109-s + 600·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 852·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.615·13-s − 0.432·37-s + 3.18·49-s − 0.307·52-s + 0.655·61-s + 3/8·64-s − 3.28·73-s − 1.23·97-s + 0.513·109-s + 4.95·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.216·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 5.04·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.927314464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.927314464\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T^{2} + p^{2} T^{4} - p^{5} T^{6} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 78 T^{2} + 4467 T^{4} - 78 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 300 T^{2} + 51318 T^{4} - 300 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + 223 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 732 T^{2} + 296822 T^{4} + 732 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 1142 T^{2} + 563947 T^{4} - 1142 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 332 T^{2} - 192746 T^{4} - 332 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 316 T^{2} + 136150 T^{4} + 316 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2814 T^{2} + 3641091 T^{4} - 2814 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 2278 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 2972 T^{2} + 7855542 T^{4} + 2972 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 38 T^{2} + 6759547 T^{4} - 38 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 5540 T^{2} + 17424838 T^{4} - 5540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4348 T^{2} + 11542294 T^{4} + 4348 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 7300 T^{2} + 36955878 T^{4} - 7300 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 10 T + 1783 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 8454 T^{2} + 40162395 T^{4} - 8454 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 1140 T^{2} + 3931158 T^{4} + 1140 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 60 T - 42 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 3396 T^{2} + 9483270 T^{4} - 3396 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 452 T^{2} + 57543334 T^{4} - 452 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 15172 T^{2} + 120195142 T^{4} + 15172 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 15 T + p^{2} T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.25311898224451134028721600875, −4.02162052150745129389383227178, −3.86230222008088436260276432673, −3.72884798723187022168560160588, −3.45494999659945590092076268123, −3.43814696206683323173630374587, −3.37369529454665674088999762339, −3.22799045532659065580972226982, −2.95206069112715233914091515470, −2.81517013130299169193278277797, −2.69691590994559465363826309398, −2.64017195533188533425716002829, −2.33402934630074449715100318700, −2.27048377859529575682372402022, −2.20172956005156837303514096819, −1.97426817794889462005536959310, −1.86384670895613313848767951690, −1.47888408853428935298548810341, −1.47151237937669116777946393792, −1.26005073290689943647780133561, −0.979098558218005957658703365437, −0.902569500989783783505077437076, −0.57864695801608518556810805214, −0.31717682405452015797761711233, −0.21020285094899872376015877251,
0.21020285094899872376015877251, 0.31717682405452015797761711233, 0.57864695801608518556810805214, 0.902569500989783783505077437076, 0.979098558218005957658703365437, 1.26005073290689943647780133561, 1.47151237937669116777946393792, 1.47888408853428935298548810341, 1.86384670895613313848767951690, 1.97426817794889462005536959310, 2.20172956005156837303514096819, 2.27048377859529575682372402022, 2.33402934630074449715100318700, 2.64017195533188533425716002829, 2.69691590994559465363826309398, 2.81517013130299169193278277797, 2.95206069112715233914091515470, 3.22799045532659065580972226982, 3.37369529454665674088999762339, 3.43814696206683323173630374587, 3.45494999659945590092076268123, 3.72884798723187022168560160588, 3.86230222008088436260276432673, 4.02162052150745129389383227178, 4.25311898224451134028721600875
Plot not available for L-functions of degree greater than 10.
