| L(s) = 1 | − 408·4-s + 9.20e4·16-s + 1.52e5·19-s + 8.26e5·31-s − 1.11e6·49-s − 1.02e6·61-s − 1.39e7·64-s − 6.21e7·76-s − 2.16e6·79-s − 2.92e7·121-s − 3.37e8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.12e8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 4.55e8·196-s + ⋯ |
| L(s) = 1 | − 3.18·4-s + 5.62·16-s + 5.09·19-s + 4.98·31-s − 1.35·49-s − 0.579·61-s − 6.63·64-s − 16.2·76-s − 0.492·79-s − 1.50·121-s − 15.8·124-s + 3.37·169-s + 4.32·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.728094647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728094647\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + 51 p^{2} T^{2} + p^{14} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 558139 T^{2} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14640342 T^{2} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 106004809 T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 156067506 T^{2} + p^{14} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 38099 T + p^{7} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 239219858 p T^{2} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 24549941382 T^{2} + p^{14} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 206583 T + p^{7} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 184168033366 T^{2} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 183368138238 T^{2} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 340538279989 T^{2} + p^{14} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 1003188351966 T^{2} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2258772175434 T^{2} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3556300225638 T^{2} + p^{14} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 257033 T + p^{7} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 1569223198421 T^{2} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 11474291422782 T^{2} + p^{14} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 20913162817294 T^{2} + p^{14} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 540084 T + p^{7} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 47583027737014 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 88313629791058 T^{2} + p^{14} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 147535831401001 T^{2} + p^{14} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.916140521631412883517455564280, −7.38617714307130966096126341601, −7.30924700186213036226142402376, −6.84855211866703691125648183835, −6.44301285977570925299810106206, −6.28070007073261803440559742345, −5.71405131631486908577195633840, −5.52335185568615732346317694590, −5.43884374783393864829796324179, −5.03605139800018217354318916040, −4.67994240247722415584626465031, −4.64902186409729075303966640241, −4.51984068061114324411897241180, −4.05458024721329081457602979249, −3.50626828510730771310022862669, −3.50028933757942039940099381442, −3.01865088311103686180199990446, −2.94029449147088754959541051701, −2.60812266762814566067078860056, −1.74044279781421666558405266244, −1.33714295411553501504429614516, −0.875872099347760941748877123988, −0.869695177443827407076687074950, −0.798472332696787847150106415909, −0.20418762579086045390811162501,
0.20418762579086045390811162501, 0.798472332696787847150106415909, 0.869695177443827407076687074950, 0.875872099347760941748877123988, 1.33714295411553501504429614516, 1.74044279781421666558405266244, 2.60812266762814566067078860056, 2.94029449147088754959541051701, 3.01865088311103686180199990446, 3.50028933757942039940099381442, 3.50626828510730771310022862669, 4.05458024721329081457602979249, 4.51984068061114324411897241180, 4.64902186409729075303966640241, 4.67994240247722415584626465031, 5.03605139800018217354318916040, 5.43884374783393864829796324179, 5.52335185568615732346317694590, 5.71405131631486908577195633840, 6.28070007073261803440559742345, 6.44301285977570925299810106206, 6.84855211866703691125648183835, 7.30924700186213036226142402376, 7.38617714307130966096126341601, 7.916140521631412883517455564280
