| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s − 3·5-s − 9·6-s + 3·7-s + 10·8-s + 6·9-s − 9·10-s + 2·11-s − 18·12-s − 2·13-s + 9·14-s + 9·15-s + 15·16-s − 8·17-s + 18·18-s − 6·19-s − 18·20-s − 9·21-s + 6·22-s + 3·23-s − 30·24-s + 6·25-s − 6·26-s − 10·27-s + 18·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s − 1.34·5-s − 3.67·6-s + 1.13·7-s + 3.53·8-s + 2·9-s − 2.84·10-s + 0.603·11-s − 5.19·12-s − 0.554·13-s + 2.40·14-s + 2.32·15-s + 15/4·16-s − 1.94·17-s + 4.24·18-s − 1.37·19-s − 4.02·20-s − 1.96·21-s + 1.27·22-s + 0.625·23-s − 6.12·24-s + 6/5·25-s − 1.17·26-s − 1.92·27-s + 3.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.42280721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.42280721\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 11 | $S_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 60 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 19 T^{2} + 19 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 188 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 16 T + 151 T^{2} - 960 T^{3} + 151 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 332 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 39 T^{2} - 60 T^{3} + 39 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 125 T^{2} - 852 T^{3} + 125 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 524 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 2 T + 75 T^{2} + 108 T^{3} + 75 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 161 T^{2} - 1096 T^{3} + 161 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 18 T + 227 T^{2} - 1900 T^{3} + 227 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 197 T^{2} - 1332 T^{3} + 197 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 2 T + 193 T^{2} + 276 T^{3} + 193 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T - 25 T^{2} - 652 T^{3} - 25 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 173 T^{2} - 128 T^{3} + 173 p T^{4} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 197 T^{2} + 1124 T^{3} + 197 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 47 T^{2} + 748 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 279 T^{2} + 380 T^{3} + 279 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.29253108572538689229701540718, −6.87202735818218635247875152603, −6.72638171282320088725571055961, −6.48555533124232623760208199573, −6.37619576812260405567399370058, −5.99809325334312046882127694042, −5.96852048835946297944909909176, −5.29956110726025084965781905538, −5.28914471373960708075962956732, −5.16092490036845980527988391723, −4.57947148124649940361072796556, −4.51155620739594390068174826934, −4.48553189354153079192766744161, −4.13042858793768706413390983999, −4.04989640627433137809718045818, −3.94513663367818623554297389662, −3.04177579686617557298860729394, −3.02564302027663880066865190222, −2.78352155938315692545058001274, −2.20180153065960298118354358128, −1.94886992326993975125391747074, −1.92280795109423715630708697016, −0.859338200676367680233421972024, −0.829292726497369960393528988290, −0.66830006314520351339843474399,
0.66830006314520351339843474399, 0.829292726497369960393528988290, 0.859338200676367680233421972024, 1.92280795109423715630708697016, 1.94886992326993975125391747074, 2.20180153065960298118354358128, 2.78352155938315692545058001274, 3.02564302027663880066865190222, 3.04177579686617557298860729394, 3.94513663367818623554297389662, 4.04989640627433137809718045818, 4.13042858793768706413390983999, 4.48553189354153079192766744161, 4.51155620739594390068174826934, 4.57947148124649940361072796556, 5.16092490036845980527988391723, 5.28914471373960708075962956732, 5.29956110726025084965781905538, 5.96852048835946297944909909176, 5.99809325334312046882127694042, 6.37619576812260405567399370058, 6.48555533124232623760208199573, 6.72638171282320088725571055961, 6.87202735818218635247875152603, 7.29253108572538689229701540718
