| L(s) = 1 | − 3·5-s + 8-s + 3·13-s − 3·19-s + 6·25-s + 27-s − 3·40-s − 9·65-s − 3·67-s − 3·71-s + 9·95-s + 3·104-s + 3·121-s − 10·125-s + 127-s + 131-s − 3·135-s + 137-s + 139-s + 149-s + 151-s − 3·152-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + ⋯ |
| L(s) = 1 | − 3·5-s + 8-s + 3·13-s − 3·19-s + 6·25-s + 27-s − 3·40-s − 9·65-s − 3·67-s − 3·71-s + 9·95-s + 3·104-s + 3·121-s − 10·125-s + 127-s + 131-s − 3·135-s + 137-s + 139-s + 149-s + 151-s − 3·152-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37595375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37595375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3543993936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3543993936\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 67 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 3 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 7 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 59 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 97 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
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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
−10.59902052273814467286363515394, −10.49055359866044824258850821501, −10.41616092914958115372164890058, −9.580150211929100173148028357842, −8.888317130052290697669565365949, −8.767496715948049424171169336994, −8.618620349712002559397034899108, −8.321120133324317664976393167418, −8.245143460775091936767199914297, −7.55562061190441576960374353751, −7.54983037810215882667931044274, −7.07016373855777459558511376582, −6.73436287797225975178421008702, −6.34802687744321409025671317649, −6.06088648943415724796174788745, −5.69510768655642927153715029441, −4.71413292823968241065386480467, −4.62358630804251537525358135336, −4.19798635186956728107007653696, −4.19374872637033989546836058558, −3.62139932911298065432906892303, −3.23953872548385471452215328700, −2.87439144930788133554962157919, −1.79478182195161881311488581269, −1.14072954590736104187152242216,
1.14072954590736104187152242216, 1.79478182195161881311488581269, 2.87439144930788133554962157919, 3.23953872548385471452215328700, 3.62139932911298065432906892303, 4.19374872637033989546836058558, 4.19798635186956728107007653696, 4.62358630804251537525358135336, 4.71413292823968241065386480467, 5.69510768655642927153715029441, 6.06088648943415724796174788745, 6.34802687744321409025671317649, 6.73436287797225975178421008702, 7.07016373855777459558511376582, 7.54983037810215882667931044274, 7.55562061190441576960374353751, 8.245143460775091936767199914297, 8.321120133324317664976393167418, 8.618620349712002559397034899108, 8.767496715948049424171169336994, 8.888317130052290697669565365949, 9.580150211929100173148028357842, 10.41616092914958115372164890058, 10.49055359866044824258850821501, 10.59902052273814467286363515394
