| L(s) = 1 | + 3-s + 3·5-s − 4·7-s − 9-s + 11-s + 3·13-s + 3·15-s + 2·17-s − 3·19-s − 4·21-s − 3·23-s + 6·25-s + 27-s + 17·29-s − 16·31-s + 33-s − 12·35-s + 9·37-s + 3·39-s + 16·41-s + 12·43-s − 3·45-s − 2·47-s − 3·49-s + 2·51-s + 17·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 1.51·7-s − 1/3·9-s + 0.301·11-s + 0.832·13-s + 0.774·15-s + 0.485·17-s − 0.688·19-s − 0.872·21-s − 0.625·23-s + 6/5·25-s + 0.192·27-s + 3.15·29-s − 2.87·31-s + 0.174·33-s − 2.02·35-s + 1.47·37-s + 0.480·39-s + 2.49·41-s + 1.82·43-s − 0.447·45-s − 0.291·47-s − 3/7·49-s + 0.280·51-s + 2.33·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.236530846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.236530846\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{2} - 4 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 19 T^{2} + p^{2} T^{3} + 19 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 26 T^{2} - 20 T^{3} + 26 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 20 T^{2} - 46 T^{3} + 20 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 23 T^{2} - 19 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 2 p T^{2} + 82 T^{3} + 2 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 17 T + 165 T^{2} - 1038 T^{3} + 165 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 16 T + 171 T^{2} + 1105 T^{3} + 171 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 83 T^{2} - 682 T^{3} + 83 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 16 T + 113 T^{2} - 611 T^{3} + 113 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 25 T^{2} + 348 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 17 T + 215 T^{2} - 1678 T^{3} + 215 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 9 T + 83 T^{2} + 242 T^{3} + 83 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 15 T + 38 T^{2} + 344 T^{3} + 38 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 23 T + 337 T^{2} + 3146 T^{3} + 337 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 137 T^{2} - 779 T^{3} + 137 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 255 T^{2} - 2012 T^{3} + 255 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 7 T + 225 T^{2} + 1158 T^{3} + 225 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 139 T^{2} - 660 T^{3} + 139 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 252 T^{2} - 1564 T^{3} + 252 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.58249309979439894775438669385, −7.31484628610046863882779818816, −6.98885716399646177510176075168, −6.60216665030186922794469857021, −6.47359414876702800443264022878, −6.42794216033113550237698967893, −6.07057533088690785455793356514, −5.80618535277995836267047461633, −5.65435869835466516265550394724, −5.49767158267033251247739097082, −4.89490687628150300771073395768, −4.87156109678869337692858834663, −4.27205250326610831372463625089, −4.16985760963618542673767513020, −3.77512075772749422283190036458, −3.69776916222438008340105519701, −3.03275105377707649895627992062, −2.95674577074131310406080466602, −2.86149912523852990880460753249, −2.22767253905240700011886841439, −2.17989638566189146789250832189, −1.85124488882691768154566926698, −1.14739843834496772085937024362, −0.75354675147234713788531671696, −0.63627712865441610685739979158,
0.63627712865441610685739979158, 0.75354675147234713788531671696, 1.14739843834496772085937024362, 1.85124488882691768154566926698, 2.17989638566189146789250832189, 2.22767253905240700011886841439, 2.86149912523852990880460753249, 2.95674577074131310406080466602, 3.03275105377707649895627992062, 3.69776916222438008340105519701, 3.77512075772749422283190036458, 4.16985760963618542673767513020, 4.27205250326610831372463625089, 4.87156109678869337692858834663, 4.89490687628150300771073395768, 5.49767158267033251247739097082, 5.65435869835466516265550394724, 5.80618535277995836267047461633, 6.07057533088690785455793356514, 6.42794216033113550237698967893, 6.47359414876702800443264022878, 6.60216665030186922794469857021, 6.98885716399646177510176075168, 7.31484628610046863882779818816, 7.58249309979439894775438669385
