| L(s) = 1 | − 8-s + 3·23-s − 27-s + 3·49-s − 3·59-s − 3·101-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 3·184-s + 191-s + 193-s + 197-s + 199-s + 211-s + 216-s + 223-s + ⋯ |
| L(s) = 1 | − 8-s + 3·23-s − 27-s + 3·49-s − 3·59-s − 3·101-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 3·184-s + 191-s + 193-s + 197-s + 199-s + 211-s + 216-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6556801873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6556801873\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 3 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 31 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 73 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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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
−9.561900889520333493608684917783, −9.339336378707880667686006703466, −9.208045727184300471542158650459, −9.136386088640053209177148978312, −8.572159039019146774274797155540, −8.350425295826187815728332777267, −8.162721893762865308279425434899, −7.52354371500980851900803976683, −7.27010152369271818712809883035, −7.07905673992222131770962254339, −6.97035821410161563802388580830, −6.20240654069126545614118117153, −6.12733772540522970572047712810, −5.83123660037314210455606434401, −5.41665914868307849860639780692, −5.10437346747888680032684664027, −4.60568097268820561108524332431, −4.54745731076551961513135339910, −3.72794771630425075235500379730, −3.64540640170479691181978455341, −3.09873853714738218617789177484, −2.65048328034380777727899129982, −2.52468541264799661452488757113, −1.62923584804268505443842366083, −1.05692993630221714098830477357,
1.05692993630221714098830477357, 1.62923584804268505443842366083, 2.52468541264799661452488757113, 2.65048328034380777727899129982, 3.09873853714738218617789177484, 3.64540640170479691181978455341, 3.72794771630425075235500379730, 4.54745731076551961513135339910, 4.60568097268820561108524332431, 5.10437346747888680032684664027, 5.41665914868307849860639780692, 5.83123660037314210455606434401, 6.12733772540522970572047712810, 6.20240654069126545614118117153, 6.97035821410161563802388580830, 7.07905673992222131770962254339, 7.27010152369271818712809883035, 7.52354371500980851900803976683, 8.162721893762865308279425434899, 8.350425295826187815728332777267, 8.572159039019146774274797155540, 9.136386088640053209177148978312, 9.208045727184300471542158650459, 9.339336378707880667686006703466, 9.561900889520333493608684917783
