| L(s) = 1 | + 2·3-s − 4·7-s + 3·9-s − 8·21-s + 2·23-s + 2·27-s + 6·29-s + 4·43-s + 4·47-s + 6·49-s − 2·61-s − 12·63-s + 4·69-s + 81-s − 2·83-s + 12·87-s − 8·101-s − 6·103-s + 8·107-s − 2·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + 8·141-s + 12·147-s + ⋯ |
| L(s) = 1 | + 2·3-s − 4·7-s + 3·9-s − 8·21-s + 2·23-s + 2·27-s + 6·29-s + 4·43-s + 4·47-s + 6·49-s − 2·61-s − 12·63-s + 4·69-s + 81-s − 2·83-s + 12·87-s − 8·101-s − 6·103-s + 8·107-s − 2·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + 8·141-s + 12·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09024714376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09024714376\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 13 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 23 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 29 | \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 37 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
| 53 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 61 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 67 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 73 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 79 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 83 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 89 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 97 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
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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
−3.74946982931191428360685380634, −3.39915574596050139354424337928, −3.37989098451508095254733239614, −3.19512390882343194869282526032, −3.19066881457474517677796049435, −3.15296087622758479518776470874, −3.00845280106997253026626527407, −2.86018878824578382324930078530, −2.70571072828552581985818144693, −2.70033166724542125463324187060, −2.64944933096877302932477146922, −2.51635316283310134377849970704, −2.45972849957572861003855248015, −2.32962611307530444105927543060, −2.20524270959923933514758482634, −2.00107106813466588740631419142, −1.95046209959734491881224361794, −1.38838509443711431897979746616, −1.31608756249831555802373810424, −1.25963153710636934195951980747, −1.20983687566452471654288351110, −1.05172027443866975910490890677, −0.939987320435293528483016134650, −0.820347105564426561840416847261, −0.04952530926415038276647925936,
0.04952530926415038276647925936, 0.820347105564426561840416847261, 0.939987320435293528483016134650, 1.05172027443866975910490890677, 1.20983687566452471654288351110, 1.25963153710636934195951980747, 1.31608756249831555802373810424, 1.38838509443711431897979746616, 1.95046209959734491881224361794, 2.00107106813466588740631419142, 2.20524270959923933514758482634, 2.32962611307530444105927543060, 2.45972849957572861003855248015, 2.51635316283310134377849970704, 2.64944933096877302932477146922, 2.70033166724542125463324187060, 2.70571072828552581985818144693, 2.86018878824578382324930078530, 3.00845280106997253026626527407, 3.15296087622758479518776470874, 3.19066881457474517677796049435, 3.19512390882343194869282526032, 3.37989098451508095254733239614, 3.39915574596050139354424337928, 3.74946982931191428360685380634
Plot not available for L-functions of degree greater than 10.
