| L(s) = 1 | − 408·4-s + 9.20e4·16-s − 1.02e5·25-s − 4.60e5·37-s + 1.51e6·43-s − 1.39e7·64-s + 4.51e6·67-s − 9.67e6·79-s + 4.20e7·100-s − 7.36e7·109-s − 7.35e7·121-s + 127-s + 131-s + 137-s + 139-s + 1.87e8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.13e7·169-s − 6.16e8·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 3.18·4-s + 5.62·16-s − 1.31·25-s − 1.49·37-s + 2.89·43-s − 6.63·64-s + 1.83·67-s − 2.20·79-s + 4.20·100-s − 5.44·109-s − 3.77·121-s + 4.76·148-s + 0.340·169-s − 9.24·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + 51 p^{2} T^{2} + p^{14} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 10298 p T^{2} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 36789042 T^{2} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 822382 p T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 802972906 T^{2} + p^{14} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 213410198 T^{2} + p^{14} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 650787066 T^{2} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 23791782618 T^{2} + p^{14} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 39723144542 T^{2} + p^{14} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 115050 T + p^{7} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 299731063238 T^{2} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 377940 T + p^{7} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 97860065566 T^{2} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2201684913066 T^{2} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4821378185638 T^{2} + p^{14} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 628993357322 T^{2} + p^{14} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 1129180 T + p^{7} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 9563153837082 T^{2} + p^{14} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 20386454766194 T^{2} + p^{14} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2418184 T + p^{7} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 37714240903414 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 24782310732058 T^{2} + p^{14} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 1328843609858 p T^{2} + p^{14} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52524025120620808239706525428, −7.34992013275571074016313898314, −6.58877581246072696482086913289, −6.54891169009845101178386315605, −6.53113639286385921109871652968, −5.86227781782058754022018265373, −5.65287570536749622445969568403, −5.57432249740223468239660605966, −5.24643237318488693965603447560, −5.15174576924225337512847789540, −4.76163207829196033778860678668, −4.57943770824485860633908225315, −4.30041172602365892812736942249, −3.99136928656639954515694624104, −3.78522447257572355305067093232, −3.65809193365394659556115392584, −3.63429593726001775019535347625, −2.88338515062538784861025856920, −2.64210888524980967374688098383, −2.39995752871751769851257304930, −2.10636269942716968377947107889, −1.38736785227826634301603615091, −1.25501643676233987297899065561, −1.08015585024506226226215926751, −0.821268738849228707155500947076, 0, 0, 0, 0,
0.821268738849228707155500947076, 1.08015585024506226226215926751, 1.25501643676233987297899065561, 1.38736785227826634301603615091, 2.10636269942716968377947107889, 2.39995752871751769851257304930, 2.64210888524980967374688098383, 2.88338515062538784861025856920, 3.63429593726001775019535347625, 3.65809193365394659556115392584, 3.78522447257572355305067093232, 3.99136928656639954515694624104, 4.30041172602365892812736942249, 4.57943770824485860633908225315, 4.76163207829196033778860678668, 5.15174576924225337512847789540, 5.24643237318488693965603447560, 5.57432249740223468239660605966, 5.65287570536749622445969568403, 5.86227781782058754022018265373, 6.53113639286385921109871652968, 6.54891169009845101178386315605, 6.58877581246072696482086913289, 7.34992013275571074016313898314, 7.52524025120620808239706525428
