| L(s) = 1 | + 4-s + 3·5-s + 7-s − 2·8-s − 11-s + 3·13-s + 3·16-s + 17-s + 6·19-s + 3·20-s + 7·23-s + 6·25-s + 28-s − 18·29-s + 6·31-s − 4·32-s + 3·35-s + 13·37-s − 6·40-s − 41-s − 44-s + 18·47-s − 4·49-s + 3·52-s − 11·53-s − 3·55-s − 2·56-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.34·5-s + 0.377·7-s − 0.707·8-s − 0.301·11-s + 0.832·13-s + 3/4·16-s + 0.242·17-s + 1.37·19-s + 0.670·20-s + 1.45·23-s + 6/5·25-s + 0.188·28-s − 3.34·29-s + 1.07·31-s − 0.707·32-s + 0.507·35-s + 2.13·37-s − 0.948·40-s − 0.156·41-s − 0.150·44-s + 2.62·47-s − 4/7·49-s + 0.416·52-s − 1.51·53-s − 0.404·55-s − 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.132789431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.132789431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 - T^{2} + p T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - T + 5 T^{2} - 30 T^{3} + 5 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + T + 17 T^{2} + 38 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 19 T^{2} + 42 T^{3} + 19 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 41 T^{2} - 164 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 53 T^{2} - 194 T^{3} + 53 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 340 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 646 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 91 T^{2} + 6 T^{3} + 91 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 128 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 18 T + 221 T^{2} - 1756 T^{3} + 221 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 11 T + 167 T^{2} + 1162 T^{3} + 167 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 129 T^{2} - 816 T^{3} + 129 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 9 T + 71 T^{2} - 254 T^{3} + 71 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 137 T^{2} - 408 T^{3} + 137 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 11 T + 237 T^{2} - 1530 T^{3} + 237 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 119 T^{2} + 532 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 5 T + 189 T^{2} - 854 T^{3} + 189 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 201 T^{2} + 1200 T^{3} + 201 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 11 T + 275 T^{2} + 1954 T^{3} + 275 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 25 T + 467 T^{2} + 5094 T^{3} + 467 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.618321240138132926479710099222, −9.139975139726218193931962025380, −9.106966165072204819750037331354, −8.855626152911665993410041039110, −8.331119901153000427704086331770, −8.046068407393169179994503054766, −7.64317386362695216035855708318, −7.47087379872489673523363566142, −7.20325079308604950179887002220, −6.64833433437964352001168506392, −6.54581875346392669929961587589, −6.01428354362857075115516384799, −5.78230755826575569186748267385, −5.54332531963769101314586777057, −5.28994245545142214570415822163, −5.10361484874060257198783181453, −4.32044664385178774185364284532, −4.08345915782330000823802932237, −3.51445313017621777717836226457, −3.15847293670893930892454280812, −2.84739340004738343312459938622, −2.33808916350099996578520064895, −1.96140697641443756608236313913, −1.25490646077372766887393526375, −0.933298169058262645999197157346,
0.933298169058262645999197157346, 1.25490646077372766887393526375, 1.96140697641443756608236313913, 2.33808916350099996578520064895, 2.84739340004738343312459938622, 3.15847293670893930892454280812, 3.51445313017621777717836226457, 4.08345915782330000823802932237, 4.32044664385178774185364284532, 5.10361484874060257198783181453, 5.28994245545142214570415822163, 5.54332531963769101314586777057, 5.78230755826575569186748267385, 6.01428354362857075115516384799, 6.54581875346392669929961587589, 6.64833433437964352001168506392, 7.20325079308604950179887002220, 7.47087379872489673523363566142, 7.64317386362695216035855708318, 8.046068407393169179994503054766, 8.331119901153000427704086331770, 8.855626152911665993410041039110, 9.106966165072204819750037331354, 9.139975139726218193931962025380, 9.618321240138132926479710099222
