| L(s) = 1 | + 16·2-s + 64·4-s − 1.02e3·8-s + 1.20e3·9-s + 8.68e3·11-s − 1.63e4·16-s + 1.92e4·18-s + 1.38e5·22-s − 2.38e4·23-s + 1.39e5·25-s + 9.11e5·29-s − 6.55e4·32-s + 7.71e4·36-s + 7.80e5·37-s − 9.26e5·43-s + 5.55e5·44-s − 3.81e5·46-s + 2.22e6·50-s + 2.54e6·53-s + 1.45e7·58-s + 7.86e5·64-s + 4.13e6·67-s + 8.12e5·71-s − 1.23e6·72-s + 1.24e7·74-s − 7.09e6·79-s + 4.78e6·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.707·8-s + 0.551·9-s + 1.96·11-s − 16-s + 0.779·18-s + 2.78·22-s − 0.408·23-s + 1.77·25-s + 6.94·29-s − 0.353·32-s + 0.275·36-s + 2.53·37-s − 1.77·43-s + 0.983·44-s − 0.578·46-s + 2.51·50-s + 2.35·53-s + 9.81·58-s + 3/8·64-s + 1.67·67-s + 0.269·71-s − 0.389·72-s + 3.58·74-s − 1.61·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(17.26082474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.26082474\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 134 p^{2} T^{2} - 41093 p^{4} T^{4} - 134 p^{16} T^{6} + p^{28} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 139002 T^{2} + 13218040379 T^{4} - 139002 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 4340 T - 651571 T^{2} - 4340 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14655638 T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 420785822 T^{2} + 8682881436814755 T^{4} + 420785822 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 1759552150 T^{2} + 2297017082786738379 T^{4} - 1759552150 p^{14} T^{6} + p^{28} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 11928 T - 3262548263 T^{2} + 11928 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 227914 T + p^{7} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 15413251010 T^{2} - \)\(51\!\cdots\!21\)\( T^{4} + 15413251010 p^{14} T^{6} + p^{28} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 390318 T + 57416263991 T^{2} - 390318 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 388577903410 T^{2} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 231524 T + p^{7} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 1007653595934 T^{2} + \)\(75\!\cdots\!87\)\( T^{4} - 1007653595934 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 1274498 T + 449634012167 T^{2} - 1274498 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 3029537098950 T^{2} + \)\(29\!\cdots\!39\)\( T^{4} - 3029537098950 p^{14} T^{6} + p^{28} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 1049495046154 T^{2} - \)\(87\!\cdots\!25\)\( T^{4} - 1049495046154 p^{14} T^{6} + p^{28} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 2067860 T - 1784666625723 T^{2} - 2067860 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 203056 T + p^{7} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 13931341537906 T^{2} + \)\(72\!\cdots\!27\)\( T^{4} - 13931341537906 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 3546040 T - 6629509304559 T^{2} + 3546040 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 29340689746154 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 65061193899858 T^{2} + \)\(22\!\cdots\!23\)\( T^{4} - 65061193899858 p^{14} T^{6} + p^{28} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 136357896895426 T^{2} + p^{14} T^{4} )^{2} \) |
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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
−8.654784640660545374470764800227, −8.559532893423464176308759904887, −8.250337411891039418346912234743, −8.059693203764948498227191875494, −7.71150380961059371058826274251, −6.81245792829042957759563436139, −6.70036949555650377460147803512, −6.66003288055408727544735585853, −6.57115539974071479610107250091, −6.22121110090525414357462520431, −5.59283351787282215906215936152, −5.19547597330696288661972958576, −4.97694229993215213016220747874, −4.69040119811945903155600633188, −4.11499791758416732723922572583, −4.10182102069499465290377776869, −4.06903724032604269832669519339, −3.16504763699991338644346345492, −2.86410157379261704248641353934, −2.68142885118405731152306829460, −2.35738106452743154582235824665, −1.18532846056890363939609935506, −1.13640405050177068160515216980, −1.10250548731128710300220704709, −0.44163063045156619768462637348,
0.44163063045156619768462637348, 1.10250548731128710300220704709, 1.13640405050177068160515216980, 1.18532846056890363939609935506, 2.35738106452743154582235824665, 2.68142885118405731152306829460, 2.86410157379261704248641353934, 3.16504763699991338644346345492, 4.06903724032604269832669519339, 4.10182102069499465290377776869, 4.11499791758416732723922572583, 4.69040119811945903155600633188, 4.97694229993215213016220747874, 5.19547597330696288661972958576, 5.59283351787282215906215936152, 6.22121110090525414357462520431, 6.57115539974071479610107250091, 6.66003288055408727544735585853, 6.70036949555650377460147803512, 6.81245792829042957759563436139, 7.71150380961059371058826274251, 8.059693203764948498227191875494, 8.250337411891039418346912234743, 8.559532893423464176308759904887, 8.654784640660545374470764800227
