| L(s) = 1 | + 4·5-s − 3·7-s − 5·9-s − 4·11-s − 2·13-s + 14·17-s + 3·23-s + 9·25-s + 2·27-s + 10·29-s + 10·31-s − 12·35-s + 22·37-s + 2·41-s − 8·43-s − 20·45-s + 2·47-s + 6·49-s − 14·53-s − 16·55-s − 20·59-s + 4·61-s + 15·63-s − 8·65-s − 8·67-s + 6·73-s + 12·77-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.13·7-s − 5/3·9-s − 1.20·11-s − 0.554·13-s + 3.39·17-s + 0.625·23-s + 9/5·25-s + 0.384·27-s + 1.85·29-s + 1.79·31-s − 2.02·35-s + 3.61·37-s + 0.312·41-s − 1.21·43-s − 2.98·45-s + 0.291·47-s + 6/7·49-s − 1.92·53-s − 2.15·55-s − 2.60·59-s + 0.512·61-s + 1.88·63-s − 0.992·65-s − 0.977·67-s + 0.702·73-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.483499393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.483499393\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + 7 T^{2} - 6 T^{3} + 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 29 T^{2} + 68 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 - 14 T + 95 T^{2} - 26 p T^{3} + 95 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 115 T^{2} - 600 T^{3} + 115 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 61 T^{2} - 10 p T^{3} + 61 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $D_{6}$ | \( 1 - 22 T + 7 p T^{2} - 52 p T^{3} + 7 p^{2} T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 + 8 T + 89 T^{2} + 384 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 121 T^{2} - 138 T^{3} + 121 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 139 T^{2} + 1140 T^{3} + 139 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 20 T + 249 T^{2} + 2130 T^{3} + 249 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T - 5 T^{2} + 542 T^{3} - 5 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 125 T^{2} + 636 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 149 T^{2} + 128 T^{3} + 149 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 87 T^{2} - 164 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 257 T^{2} - 1796 T^{3} + 257 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 257 T^{2} + 1312 T^{3} + 257 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 275 T^{2} - 2570 T^{3} + 275 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 11 T^{2} + 1182 T^{3} + 11 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.600796437254049871156483409086, −8.132151362952353897726325371997, −7.996639914391978337776264142791, −7.974070903387537788894964281842, −7.53450777321027428060381354201, −7.38844797981363239535880648192, −6.66313753859540812405418132620, −6.37431585162889513737134393026, −6.34499045003953149734510857282, −6.14207150058791477977948048025, −5.66700884686465149200922430995, −5.63249663012454666436272289381, −5.32494653294355784530932110786, −4.84838783136515717940890214095, −4.67015602366288669549549167989, −4.51421301579866151638100734588, −3.52171008825010458491114133934, −3.37068484736954639928906788078, −2.91952600621088075128233395634, −2.75740846620274694650110440378, −2.74671277202934876610319239531, −2.24646305852503286992868834407, −1.44785785173240794449489535000, −1.00076015132035651466054313744, −0.60473451846956100797288405238,
0.60473451846956100797288405238, 1.00076015132035651466054313744, 1.44785785173240794449489535000, 2.24646305852503286992868834407, 2.74671277202934876610319239531, 2.75740846620274694650110440378, 2.91952600621088075128233395634, 3.37068484736954639928906788078, 3.52171008825010458491114133934, 4.51421301579866151638100734588, 4.67015602366288669549549167989, 4.84838783136515717940890214095, 5.32494653294355784530932110786, 5.63249663012454666436272289381, 5.66700884686465149200922430995, 6.14207150058791477977948048025, 6.34499045003953149734510857282, 6.37431585162889513737134393026, 6.66313753859540812405418132620, 7.38844797981363239535880648192, 7.53450777321027428060381354201, 7.974070903387537788894964281842, 7.996639914391978337776264142791, 8.132151362952353897726325371997, 8.600796437254049871156483409086
