| L(s) = 1 | + 2·3-s + 9-s − 2·11-s + 2·17-s − 6·19-s − 2·23-s − 2·27-s + 20·29-s − 2·31-s − 4·33-s − 4·37-s − 12·41-s − 8·43-s + 12·47-s − 7·49-s + 4·51-s − 4·53-s − 12·57-s − 6·59-s − 16·61-s − 4·67-s − 4·69-s + 44·71-s + 6·79-s − 4·81-s + 12·83-s + 40·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.603·11-s + 0.485·17-s − 1.37·19-s − 0.417·23-s − 0.384·27-s + 3.71·29-s − 0.359·31-s − 0.696·33-s − 0.657·37-s − 1.87·41-s − 1.21·43-s + 1.75·47-s − 49-s + 0.560·51-s − 0.549·53-s − 1.58·57-s − 0.781·59-s − 2.04·61-s − 0.488·67-s − 0.481·69-s + 5.22·71-s + 0.675·79-s − 4/9·81-s + 1.31·83-s + 4.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03391035509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03391035509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 11 | $D_4\times C_2$ | \( 1 + 2 T - 12 T^{2} - 12 T^{3} + 91 T^{4} - 12 p T^{5} - 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 12 p T^{5} - 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 6 p T^{3} - 36 p T^{4} - 6 p^{2} T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 12 p T^{5} - 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 10 T + 76 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 2 T - p T^{2} - 54 T^{3} + 140 T^{4} - 54 p T^{5} - p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 55 T^{2} - 12 T^{3} + 3080 T^{4} - 12 p T^{5} - 55 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T - 66 T^{2} - 96 T^{3} + 3067 T^{4} - 96 p T^{5} - 66 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 28 T^{2} - 324 T^{3} - 1509 T^{4} - 324 p T^{5} - 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4 T - 115 T^{2} - 12 T^{3} + 11600 T^{4} - 12 p T^{5} - 115 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 22 T + 256 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 29 T^{2} - 4488 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 103 T^{2} + 114 T^{3} + 10236 T^{4} + 114 p T^{5} - 103 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 2 T + 348 T^{3} - 8261 T^{4} + 348 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 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
−6.48222300275815638737762001224, −6.28548610596610543181639363189, −6.13548369069432546898939869506, −6.08714186704864156519377879029, −5.52791281237267417229014258751, −5.36061286353747950844992846306, −5.01383023649767751770725840721, −4.95797595604611718842242042495, −4.89624815168084266509231324679, −4.62009084358237007655628032694, −4.21503280202259462139539935015, −4.15769982052311540891757990430, −3.60032505420382364134945238221, −3.58389721269136545931106571507, −3.52710776362416628698887256813, −3.02155007653344811491216569505, −2.92577324543165517348446567507, −2.66060236984952338554894876581, −2.30814256297129907361802393111, −2.18889918748144899683826263925, −1.93771324536031812313476644248, −1.51471765055407136361894522842, −1.11398651682934146680990877536, −0.910013020525929562130353992357, −0.02713840032618006189778543828,
0.02713840032618006189778543828, 0.910013020525929562130353992357, 1.11398651682934146680990877536, 1.51471765055407136361894522842, 1.93771324536031812313476644248, 2.18889918748144899683826263925, 2.30814256297129907361802393111, 2.66060236984952338554894876581, 2.92577324543165517348446567507, 3.02155007653344811491216569505, 3.52710776362416628698887256813, 3.58389721269136545931106571507, 3.60032505420382364134945238221, 4.15769982052311540891757990430, 4.21503280202259462139539935015, 4.62009084358237007655628032694, 4.89624815168084266509231324679, 4.95797595604611718842242042495, 5.01383023649767751770725840721, 5.36061286353747950844992846306, 5.52791281237267417229014258751, 6.08714186704864156519377879029, 6.13548369069432546898939869506, 6.28548610596610543181639363189, 6.48222300275815638737762001224
