| L(s) = 1 | + 568·25-s − 908·49-s + 3.40e3·73-s + 6.39e3·97-s + 8.20e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.57e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4.54·25-s − 2.64·49-s + 5.45·73-s + 6.69·97-s + 6.16·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 7.16·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(44.66222806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(44.66222806\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 142 T^{2} + p^{6} T^{4} )^{4} \) |
| 7 | \( ( 1 + 227 T^{2} + p^{6} T^{4} )^{4} \) |
| 11 | \( ( 1 - 2050 T^{2} + p^{6} T^{4} )^{4} \) |
| 13 | \( ( 1 - 3935 T^{2} + p^{6} T^{4} )^{4} \) |
| 17 | \( ( 1 - 322 p T^{2} + p^{6} T^{4} )^{4} \) |
| 19 | \( ( 1 - 13669 T^{2} + p^{6} T^{4} )^{4} \) |
| 23 | \( ( 1 + 11266 T^{2} + p^{6} T^{4} )^{4} \) |
| 29 | \( ( 1 + 24230 T^{2} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 + 13682 T^{2} + p^{6} T^{4} )^{4} \) |
| 37 | \( ( 1 - 45767 T^{2} + p^{6} T^{4} )^{4} \) |
| 41 | \( ( 1 - 18830 T^{2} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 - 150550 T^{2} + p^{6} T^{4} )^{4} \) |
| 47 | \( ( 1 + 194578 T^{2} + p^{6} T^{4} )^{4} \) |
| 53 | \( ( 1 - 297322 T^{2} + p^{6} T^{4} )^{4} \) |
| 59 | \( ( 1 - 380770 T^{2} + p^{6} T^{4} )^{4} \) |
| 61 | \( ( 1 - 251543 T^{2} + p^{6} T^{4} )^{4} \) |
| 67 | \( ( 1 - 476917 T^{2} + p^{6} T^{4} )^{4} \) |
| 71 | \( ( 1 + 92014 T^{2} + p^{6} T^{4} )^{4} \) |
| 73 | \( ( 1 - 425 T + p^{3} T^{2} )^{8} \) |
| 79 | \( ( 1 - 721861 T^{2} + p^{6} T^{4} )^{4} \) |
| 83 | \( ( 1 - 791062 T^{2} + p^{6} T^{4} )^{4} \) |
| 89 | \( ( 1 + 1404430 T^{2} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 - 799 T + p^{3} T^{2} )^{8} \) |
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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
−3.55134342412827757762735936777, −3.20523770368075345952610976940, −3.17462469383289070566744192680, −3.14938342042178998345031530558, −3.10562693431135813340551063842, −2.89867117372510394655084949261, −2.88258178676077067411798557267, −2.72401336168693127828041286466, −2.64583575798987003867496119028, −2.22211870050629610423795180697, −2.18026715244759480572893873080, −2.04151487096927540978738302701, −2.00545513105599137784198727688, −1.78298166232280878960155296582, −1.65922213300040722744285006228, −1.65314388110765632780005883399, −1.50620414939785917410177494999, −1.16743160874102306812327924721, −0.872086490516631685917743095055, −0.70153120918985975560966998515, −0.68730973755729212158423443148, −0.64799529227701127738520215540, −0.59680120133221078413415178801, −0.56173951121881049665300707345, −0.19193238197150231436102342985,
0.19193238197150231436102342985, 0.56173951121881049665300707345, 0.59680120133221078413415178801, 0.64799529227701127738520215540, 0.68730973755729212158423443148, 0.70153120918985975560966998515, 0.872086490516631685917743095055, 1.16743160874102306812327924721, 1.50620414939785917410177494999, 1.65314388110765632780005883399, 1.65922213300040722744285006228, 1.78298166232280878960155296582, 2.00545513105599137784198727688, 2.04151487096927540978738302701, 2.18026715244759480572893873080, 2.22211870050629610423795180697, 2.64583575798987003867496119028, 2.72401336168693127828041286466, 2.88258178676077067411798557267, 2.89867117372510394655084949261, 3.10562693431135813340551063842, 3.14938342042178998345031530558, 3.17462469383289070566744192680, 3.20523770368075345952610976940, 3.55134342412827757762735936777
Plot not available for L-functions of degree greater than 10.
