| L(s) = 1 | + 3-s − 2·5-s − 9-s − 3·11-s − 8·13-s − 2·15-s − 9·17-s + 3·19-s + 4·23-s − 2·25-s − 5·27-s + 11·29-s + 5·31-s − 3·33-s + 4·37-s − 8·39-s − 11·41-s − 4·43-s + 2·45-s + 2·47-s − 9·51-s + 9·53-s + 6·55-s + 3·57-s + 12·59-s + 2·61-s + 16·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1/3·9-s − 0.904·11-s − 2.21·13-s − 0.516·15-s − 2.18·17-s + 0.688·19-s + 0.834·23-s − 2/5·25-s − 0.962·27-s + 2.04·29-s + 0.898·31-s − 0.522·33-s + 0.657·37-s − 1.28·39-s − 1.71·41-s − 0.609·43-s + 0.298·45-s + 0.291·47-s − 1.26·51-s + 1.23·53-s + 0.809·55-s + 0.397·57-s + 1.56·59-s + 0.256·61-s + 1.98·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9999869964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9999869964\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{2} + 2 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 6 T^{2} + 6 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 26 T^{2} + 46 T^{3} + 26 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 50 T^{2} + 192 T^{3} + 50 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 9 T + 4 p T^{2} + 292 T^{3} + 4 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 64 T^{2} - 180 T^{3} + 64 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 11 T + 102 T^{2} - 636 T^{3} + 102 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 5 T + 76 T^{2} - 314 T^{3} + 76 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 92 T^{2} - 246 T^{3} + 92 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 11 T + 110 T^{2} + 622 T^{3} + 110 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 37 T^{2} + 376 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 66 T^{2} - 372 T^{3} + 66 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 120 T^{2} - 632 T^{3} + 120 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 132 T^{2} - 1308 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 26 T^{2} - 42 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 9 T + 218 T^{2} + 1192 T^{3} + 218 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 158 T^{2} - 58 T^{3} + 158 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 21 T + 302 T^{2} - 2764 T^{3} + 302 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} + 68 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 7 T + 204 T^{2} - 1062 T^{3} + 204 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 335 T^{2} - 3268 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 28 T + 330 T^{2} + 2896 T^{3} + 330 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.61816682896335726648521725943, −7.42228490380432989464561480579, −6.96574417639384905792170847121, −6.96105404906214256907110247291, −6.51032919360125665819937545831, −6.47041209633109383001378352288, −6.29461274622035733698704405128, −5.43759133939516394622532790395, −5.40207560207854206368336541064, −5.39432915415220208762407129109, −4.88330718055391867467655135037, −4.73520529111362876360908983046, −4.48928696800313816782977062905, −4.21607902051530209116242952510, −3.84027038882360792931192399519, −3.68092717547008909006322633128, −3.07700190643994908161055391844, −3.01430408480641481016702309557, −2.56478679926460993918607264808, −2.45626779252593197297231129562, −2.23442062372540720502705577906, −1.83069808231681772991463293395, −1.19378433966741929290472365821, −0.61449861086728187076872324367, −0.26098274583688246480145691657,
0.26098274583688246480145691657, 0.61449861086728187076872324367, 1.19378433966741929290472365821, 1.83069808231681772991463293395, 2.23442062372540720502705577906, 2.45626779252593197297231129562, 2.56478679926460993918607264808, 3.01430408480641481016702309557, 3.07700190643994908161055391844, 3.68092717547008909006322633128, 3.84027038882360792931192399519, 4.21607902051530209116242952510, 4.48928696800313816782977062905, 4.73520529111362876360908983046, 4.88330718055391867467655135037, 5.39432915415220208762407129109, 5.40207560207854206368336541064, 5.43759133939516394622532790395, 6.29461274622035733698704405128, 6.47041209633109383001378352288, 6.51032919360125665819937545831, 6.96105404906214256907110247291, 6.96574417639384905792170847121, 7.42228490380432989464561480579, 7.61816682896335726648521725943
