| L(s) = 1 | + 2-s − 6·4-s + 4·7-s − 14·8-s − 52·11-s − 2·13-s + 4·14-s − 29·16-s + 64·17-s − 46·19-s − 52·22-s − 90·23-s − 2·26-s − 24·28-s − 470·29-s − 262·31-s + 11·32-s + 64·34-s − 542·37-s − 46·38-s − 698·41-s + 142·43-s + 312·44-s − 90·46-s − 542·47-s − 288·49-s + 12·52-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 3/4·4-s + 0.215·7-s − 0.618·8-s − 1.42·11-s − 0.0426·13-s + 0.0763·14-s − 0.453·16-s + 0.913·17-s − 0.555·19-s − 0.503·22-s − 0.815·23-s − 0.0150·26-s − 0.161·28-s − 3.00·29-s − 1.51·31-s + 0.0607·32-s + 0.322·34-s − 2.40·37-s − 0.196·38-s − 2.65·41-s + 0.503·43-s + 1.06·44-s − 0.288·46-s − 1.68·47-s − 0.839·49-s + 0.0320·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + 7 T^{2} + T^{3} + 7 p^{3} T^{4} + p^{3} T^{5} + 7 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 4 T + 304 T^{2} + 1284 T^{3} + 135246 T^{4} + 1284 p^{3} T^{5} + 304 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 52 T + 5362 T^{2} + 190856 T^{3} + 10824347 T^{4} + 190856 p^{3} T^{5} + 5362 p^{6} T^{6} + 52 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2 T + 6472 T^{2} - 26588 T^{3} + 18910625 T^{4} - 26588 p^{3} T^{5} + 6472 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 64 T + 7768 T^{2} - 426152 T^{3} + 51316286 T^{4} - 426152 p^{3} T^{5} + 7768 p^{6} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 46 T + 12952 T^{2} - 258226 T^{3} + 60569102 T^{4} - 258226 p^{3} T^{5} + 12952 p^{6} T^{6} + 46 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 90 T + 29876 T^{2} + 2650320 T^{3} + 431192517 T^{4} + 2650320 p^{3} T^{5} + 29876 p^{6} T^{6} + 90 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 470 T + 146368 T^{2} + 29467930 T^{3} + 5173396862 T^{4} + 29467930 p^{3} T^{5} + 146368 p^{6} T^{6} + 470 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 262 T + 81328 T^{2} + 20833878 T^{3} + 3324951870 T^{4} + 20833878 p^{3} T^{5} + 81328 p^{6} T^{6} + 262 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 542 T + 178048 T^{2} + 40244348 T^{3} + 8210386345 T^{4} + 40244348 p^{3} T^{5} + 178048 p^{6} T^{6} + 542 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 698 T + 326752 T^{2} + 119256214 T^{3} + 36621841502 T^{4} + 119256214 p^{3} T^{5} + 326752 p^{6} T^{6} + 698 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 142 T + 58792 T^{2} + 17559058 T^{3} + 758268590 T^{4} + 17559058 p^{3} T^{5} + 58792 p^{6} T^{6} - 142 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 542 T + 410020 T^{2} + 135158128 T^{3} + 59790085157 T^{4} + 135158128 p^{3} T^{5} + 410020 p^{6} T^{6} + 542 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 910 T + 830824 T^{2} - 425422754 T^{3} + 203391921422 T^{4} - 425422754 p^{3} T^{5} + 830824 p^{6} T^{6} - 910 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 100 T + 385378 T^{2} + 22341320 T^{3} + 113461174667 T^{4} + 22341320 p^{3} T^{5} + 385378 p^{6} T^{6} + 100 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 74 T + 666220 T^{2} - 43867444 T^{3} + 213354562325 T^{4} - 43867444 p^{3} T^{5} + 666220 p^{6} T^{6} - 74 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 928 T + 810160 T^{2} - 498112392 T^{3} + 334787409822 T^{4} - 498112392 p^{3} T^{5} + 810160 p^{6} T^{6} - 928 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 1622 T + 1718212 T^{2} + 1368521056 T^{3} + 890196472997 T^{4} + 1368521056 p^{3} T^{5} + 1718212 p^{6} T^{6} + 1622 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 536 T + 1056940 T^{2} + 643713256 T^{3} + 536633970182 T^{4} + 643713256 p^{3} T^{5} + 1056940 p^{6} T^{6} + 536 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 508 T + 1879072 T^{2} + 714753876 T^{3} + 1370663579118 T^{4} + 714753876 p^{3} T^{5} + 1879072 p^{6} T^{6} + 508 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1524 T + 1379804 T^{2} - 207707508 T^{3} - 51355824906 T^{4} - 207707508 p^{3} T^{5} + 1379804 p^{6} T^{6} - 1524 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 756 T + 915728 T^{2} + 714592692 T^{3} + 968859180942 T^{4} + 714592692 p^{3} T^{5} + 915728 p^{6} T^{6} + 756 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 892 T + 2662138 T^{2} - 1851314128 T^{3} + 3178868238115 T^{4} - 1851314128 p^{3} T^{5} + 2662138 p^{6} T^{6} - 892 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−7.50432096779957889607747771894, −7.42408484471973813201674982727, −7.07881779161021608294264061851, −7.02864968889576507940921534958, −6.74648180662757080955229361966, −6.33604589010960908119855121652, −6.17555228664415708446372483124, −5.82807314272545396758368900462, −5.52639896201884641943332528324, −5.33419839903511723584942004845, −5.20495991522191636241376469745, −5.10701033563621365934267794233, −4.96043870629066421312157800751, −4.29786771921469309178142500520, −4.11178819592755729691750946418, −3.87076781099840177216705696126, −3.73455639905578089459081141566, −3.42144522413533298571942710800, −3.05069243580850098983302965566, −2.90948434843223767084979142461, −2.26837266469943916964161671937, −2.15127974809302038785940115539, −1.82281898029676283071440272758, −1.36847008875610006275517218067, −1.26709928904406110050944629086, 0, 0, 0, 0,
1.26709928904406110050944629086, 1.36847008875610006275517218067, 1.82281898029676283071440272758, 2.15127974809302038785940115539, 2.26837266469943916964161671937, 2.90948434843223767084979142461, 3.05069243580850098983302965566, 3.42144522413533298571942710800, 3.73455639905578089459081141566, 3.87076781099840177216705696126, 4.11178819592755729691750946418, 4.29786771921469309178142500520, 4.96043870629066421312157800751, 5.10701033563621365934267794233, 5.20495991522191636241376469745, 5.33419839903511723584942004845, 5.52639896201884641943332528324, 5.82807314272545396758368900462, 6.17555228664415708446372483124, 6.33604589010960908119855121652, 6.74648180662757080955229361966, 7.02864968889576507940921534958, 7.07881779161021608294264061851, 7.42408484471973813201674982727, 7.50432096779957889607747771894
