| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 4·7-s − 2·8-s + 9-s − 4·10-s + 6·12-s − 2·13-s + 8·14-s + 4·15-s + 8·17-s − 2·18-s + 16·19-s + 6·20-s − 8·21-s − 4·23-s − 4·24-s + 25-s + 4·26-s − 2·27-s − 12·28-s − 8·29-s − 8·30-s − 20·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 1.51·7-s − 0.707·8-s + 1/3·9-s − 1.26·10-s + 1.73·12-s − 0.554·13-s + 2.13·14-s + 1.03·15-s + 1.94·17-s − 0.471·18-s + 3.67·19-s + 1.34·20-s − 1.74·21-s − 0.834·23-s − 0.816·24-s + 1/5·25-s + 0.784·26-s − 0.384·27-s − 2.26·28-s − 1.48·29-s − 1.46·30-s − 3.59·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.301225439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301225439\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T - 15 T^{2} - 14 T^{3} + 140 T^{4} - 14 p T^{5} - 15 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 22 T^{2} - 64 T^{3} + 387 T^{4} - 64 p T^{5} + 22 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 4 T - 26 T^{2} - 16 T^{3} + 867 T^{4} - 16 p T^{5} - 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 8 T + 22 T^{2} - 128 T^{3} - 933 T^{4} - 128 p T^{5} + 22 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 74 T^{2} + 3795 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T - 3 T^{2} + 170 T^{3} + 4460 T^{4} + 170 p T^{5} - 3 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 22 T^{2} - 336 T^{3} + 5907 T^{4} - 336 p T^{5} + 22 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T + 33 T^{2} - 714 T^{3} - 5908 T^{4} - 714 p T^{5} + 33 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T - 35 T^{2} + 378 T^{3} - 1860 T^{4} + 378 p T^{5} - 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 2 T - 71 T^{2} - 142 T^{3} + 4 T^{4} - 142 p T^{5} - 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 117 T^{2} - 882 T^{3} + 11012 T^{4} - 882 p T^{5} + 117 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 58 T^{2} - 448 T^{3} + 1267 T^{4} - 448 p T^{5} - 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.697558677410357641512976589855, −8.624376751737401864417724159078, −7.918550472571814294428617188433, −7.83511083891169464320067762999, −7.67873442247091151783772898953, −7.25369428120164197946998489691, −7.16736792364550433085930013339, −7.03783851096672845128185869822, −7.02607939682509670102950005724, −6.13782312693259007017630299683, −5.93176973212319772889184512126, −5.71716655994584507347787726499, −5.61132187096502854683716130706, −5.18213437904242541020277990687, −5.08596275089432113087427880404, −4.43315563257410613202038051105, −3.83022655100526189477527971678, −3.38309768630268570955118127202, −3.33620189312162287666657318407, −3.13314731433638830095454856736, −3.00517892019937582283499914898, −1.85927664038646741948367273824, −1.83446124223657995218546554359, −1.77082502184961863227353430648, −0.62688325020116033969017506592,
0.62688325020116033969017506592, 1.77082502184961863227353430648, 1.83446124223657995218546554359, 1.85927664038646741948367273824, 3.00517892019937582283499914898, 3.13314731433638830095454856736, 3.33620189312162287666657318407, 3.38309768630268570955118127202, 3.83022655100526189477527971678, 4.43315563257410613202038051105, 5.08596275089432113087427880404, 5.18213437904242541020277990687, 5.61132187096502854683716130706, 5.71716655994584507347787726499, 5.93176973212319772889184512126, 6.13782312693259007017630299683, 7.02607939682509670102950005724, 7.03783851096672845128185869822, 7.16736792364550433085930013339, 7.25369428120164197946998489691, 7.67873442247091151783772898953, 7.83511083891169464320067762999, 7.918550472571814294428617188433, 8.624376751737401864417724159078, 8.697558677410357641512976589855
