| L(s) = 1 | + 5·5-s + 3·7-s − 2·11-s + 3·13-s + 12·17-s + 3·19-s + 8·25-s − 29-s + 3·31-s + 15·35-s − 3·37-s + 22·41-s + 3·43-s − 9·47-s + 6·49-s + 18·53-s − 10·55-s − 9·59-s − 6·61-s + 15·65-s + 9·71-s + 3·73-s − 6·77-s − 15·79-s − 12·83-s + 60·85-s + 2·89-s + ⋯ |
| L(s) = 1 | + 2.23·5-s + 1.13·7-s − 0.603·11-s + 0.832·13-s + 2.91·17-s + 0.688·19-s + 8/5·25-s − 0.185·29-s + 0.538·31-s + 2.53·35-s − 0.493·37-s + 3.43·41-s + 0.457·43-s − 1.31·47-s + 6/7·49-s + 2.47·53-s − 1.34·55-s − 1.17·59-s − 0.768·61-s + 1.86·65-s + 1.06·71-s + 0.351·73-s − 0.683·77-s − 1.68·79-s − 1.31·83-s + 6.50·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.62933508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.62933508\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - p T + 17 T^{2} - 39 T^{3} + 17 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 14 T^{2} - 3 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 - 12 T + 90 T^{2} - 435 T^{3} + 90 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 107 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 36 T^{2} - 9 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + T + 83 T^{2} + 57 T^{3} + 83 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 213 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 22 T + 278 T^{2} - 2157 T^{3} + 278 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 3 T + 63 T^{2} - 379 T^{3} + 63 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 87 T^{2} + 657 T^{3} + 87 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 18 T + 234 T^{2} - 1917 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 9 T + 171 T^{2} + 999 T^{3} + 171 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 162 T^{2} + 665 T^{3} + 162 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T^{2} - 683 T^{3} - 6 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 681 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 15 T + 189 T^{2} + 1601 T^{3} + 189 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 288 T^{2} + 2019 T^{3} + 288 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 116 T^{2} - 735 T^{3} + 116 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 3 T + 177 T^{2} + 21 T^{3} + 177 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.00062024358702394383049678718, −6.24016514071145143069694463666, −6.23286477164653265634284251459, −6.17795154977534849589422768204, −5.77256933548015013723448711351, −5.72626227360949152012655161868, −5.60282606598749572994548976601, −5.09842492159626030127868293790, −5.09213530228282607334483455362, −5.06238144075881043186050596758, −4.48474285735323855261820681894, −4.18702204619048661703442614580, −4.07139614428624072470711986228, −3.59912172983727882308654735203, −3.50676426229852747374846894089, −3.11088610112192956194849862180, −2.69449569216128473217024340306, −2.61734756217443678363375547628, −2.49292613774783915446757985674, −1.77454490816872335353958693450, −1.68632252229957730632698970736, −1.66129338501147757300177426639, −0.988222803437643961103715042710, −0.958886695230067146015558401349, −0.58135057260803510086558959690,
0.58135057260803510086558959690, 0.958886695230067146015558401349, 0.988222803437643961103715042710, 1.66129338501147757300177426639, 1.68632252229957730632698970736, 1.77454490816872335353958693450, 2.49292613774783915446757985674, 2.61734756217443678363375547628, 2.69449569216128473217024340306, 3.11088610112192956194849862180, 3.50676426229852747374846894089, 3.59912172983727882308654735203, 4.07139614428624072470711986228, 4.18702204619048661703442614580, 4.48474285735323855261820681894, 5.06238144075881043186050596758, 5.09213530228282607334483455362, 5.09842492159626030127868293790, 5.60282606598749572994548976601, 5.72626227360949152012655161868, 5.77256933548015013723448711351, 6.17795154977534849589422768204, 6.23286477164653265634284251459, 6.24016514071145143069694463666, 7.00062024358702394383049678718
