| L(s) = 1 | + 12·9-s − 96·29-s + 336·41-s − 104·49-s + 688·61-s + 90·81-s + 528·89-s + 1.44e3·101-s + 592·109-s + 680·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 568·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 3.31·29-s + 8.19·41-s − 2.12·49-s + 11.2·61-s + 10/9·81-s + 5.93·89-s + 14.2·101-s + 5.43·109-s + 5.61·121-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.36·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(71.28618108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(71.28618108\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| good | 7 | \( ( 1 + 52 T^{2} + 2598 T^{4} + 52 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 340 T^{2} + 55302 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 284 T^{2} + 65766 T^{4} - 284 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 508 T^{2} + 127878 T^{4} - 508 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 580 T^{2} + 160422 T^{4} - 580 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 1540 T^{2} + 1106502 T^{4} + 1540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 24 T + 206 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 2980 T^{2} + 3882822 T^{4} - 2980 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 1916 T^{2} + 3514086 T^{4} - 1916 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 42 T + p^{2} T^{2} )^{8} \) |
| 43 | \( ( 1 - 380 T^{2} + 6136422 T^{4} - 380 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 6532 T^{2} + 19688838 T^{4} + 6532 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 3388 T^{2} + 3720678 T^{4} - 3388 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 10900 T^{2} + 53588742 T^{4} - 10900 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 172 T + 14118 T^{2} - 172 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 + 12100 T^{2} + 70269222 T^{4} + 12100 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 3452 T^{2} + 51544518 T^{4} + 3452 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 7388 T^{2} + 57896838 T^{4} - 7388 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 12242 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 9412 T^{2} + 57343398 T^{4} + 9412 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 132 T + 19478 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 17084 T^{2} + 164822406 T^{4} - 17084 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.13838192536886854632865923562, −3.57875715665626668660962785135, −3.52079113843882603371862852513, −3.50628833272835279935256085925, −3.44104180156542655116821696282, −3.42208880469187234205925485857, −3.38398589855881022084646574501, −3.15969665932023034237281017384, −3.01061123119158719153214195010, −2.51425110271615434878655589382, −2.27326342391918581576530394389, −2.26473575028384767057259986298, −2.20581351643953863492407900216, −2.10315033947334876446600885785, −2.07197470754747796019278364496, −2.06233477605247603305605896199, −2.02496732490815125657345648347, −1.52412609262722556512247975500, −1.16857008345063122742915696523, −0.828523876769606723646833993804, −0.818403176912807703353813934311, −0.795737024737941973730269839621, −0.60560450914674636471261472714, −0.56283902906985647734336586758, −0.54467475519098005421460318573,
0.54467475519098005421460318573, 0.56283902906985647734336586758, 0.60560450914674636471261472714, 0.795737024737941973730269839621, 0.818403176912807703353813934311, 0.828523876769606723646833993804, 1.16857008345063122742915696523, 1.52412609262722556512247975500, 2.02496732490815125657345648347, 2.06233477605247603305605896199, 2.07197470754747796019278364496, 2.10315033947334876446600885785, 2.20581351643953863492407900216, 2.26473575028384767057259986298, 2.27326342391918581576530394389, 2.51425110271615434878655589382, 3.01061123119158719153214195010, 3.15969665932023034237281017384, 3.38398589855881022084646574501, 3.42208880469187234205925485857, 3.44104180156542655116821696282, 3.50628833272835279935256085925, 3.52079113843882603371862852513, 3.57875715665626668660962785135, 4.13838192536886854632865923562
Plot not available for L-functions of degree greater than 10.
