| L(s) = 1 | − 4·2-s + 4·4-s + 16·8-s + 80·11-s − 64·16-s − 320·22-s + 136·23-s + 200·25-s − 440·29-s + 64·32-s + 40·37-s − 1.36e3·43-s + 320·44-s − 544·46-s − 800·50-s + 1.25e3·53-s + 1.76e3·58-s + 192·64-s − 1.08e3·67-s + 1.68e3·71-s − 160·74-s + 1.52e3·79-s + 5.44e3·86-s + 1.28e3·88-s + 544·92-s + 800·100-s − 5.02e3·106-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.707·8-s + 2.19·11-s − 16-s − 3.10·22-s + 1.23·23-s + 8/5·25-s − 2.81·29-s + 0.353·32-s + 0.177·37-s − 4.82·43-s + 1.09·44-s − 1.74·46-s − 2.26·50-s + 3.25·53-s + 3.98·58-s + 3/8·64-s − 1.96·67-s + 2.80·71-s − 0.251·74-s + 2.16·79-s + 6.82·86-s + 1.55·88-s + 0.616·92-s + 4/5·100-s − 4.60·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01620683825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01620683825\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - 8 p^{2} T^{2} + 39 p^{4} T^{4} - 8 p^{8} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 40 T + 269 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 344 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 9824 T^{2} + 72373407 T^{4} - 9824 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 13590 T^{2} + 137642219 T^{4} - 13590 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 68 T - 7543 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 110 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 45470 T^{2} + 1180017219 T^{4} - 45470 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 20 T - 50253 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 135392 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 340 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 199454 T^{2} + 29002682787 T^{4} - 199454 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 628 T + 245507 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 358042 T^{2} + 86013540123 T^{4} + 358042 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 388440 T^{2} + 99365259239 T^{4} + 388440 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 540 T - 9163 T^{2} + 540 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 420 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 693984 T^{2} + 330279565967 T^{4} - 693984 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 760 T + 84561 T^{2} - 760 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 251126 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 81488 T^{2} - 490340996817 T^{4} - 81488 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 1573296 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−7.04063565500013781467642204196, −6.65446219698949006125149972279, −6.46324608941313022423697520897, −6.30338605863651200564953700339, −6.26979557701873366801873111207, −5.58278123075682828782450882202, −5.42146974557432499100216895794, −5.30112705625294879631106606089, −5.10284611579858030000841392820, −4.66166331138547206186628767969, −4.50815662897138438606551624939, −4.32713596434689211206408774139, −3.80784631563840860649979271410, −3.64697950930605026297558246514, −3.50897636941868257749518980698, −3.24842772335906478978031680372, −2.98708632948703489660925369431, −2.40639646026928557675426577975, −1.92830884183618383195272699126, −1.90665733730028605449084943467, −1.69773132791504263229860553539, −1.05516544911179847314380282498, −0.908654274835398097109291404080, −0.78554096857725900419915653388, −0.02171972813948418476160604647,
0.02171972813948418476160604647, 0.78554096857725900419915653388, 0.908654274835398097109291404080, 1.05516544911179847314380282498, 1.69773132791504263229860553539, 1.90665733730028605449084943467, 1.92830884183618383195272699126, 2.40639646026928557675426577975, 2.98708632948703489660925369431, 3.24842772335906478978031680372, 3.50897636941868257749518980698, 3.64697950930605026297558246514, 3.80784631563840860649979271410, 4.32713596434689211206408774139, 4.50815662897138438606551624939, 4.66166331138547206186628767969, 5.10284611579858030000841392820, 5.30112705625294879631106606089, 5.42146974557432499100216895794, 5.58278123075682828782450882202, 6.26979557701873366801873111207, 6.30338605863651200564953700339, 6.46324608941313022423697520897, 6.65446219698949006125149972279, 7.04063565500013781467642204196
