| L(s) = 1 | + 2-s + 3·3-s − 2·4-s + 2·5-s + 3·6-s − 3·7-s − 2·8-s + 6·9-s + 2·10-s − 10·11-s − 6·12-s − 3·14-s + 6·15-s + 16-s + 6·18-s − 8·19-s − 4·20-s − 9·21-s − 10·22-s − 2·23-s − 6·24-s − 3·25-s + 10·27-s + 6·28-s − 2·29-s + 6·30-s − 12·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 4-s + 0.894·5-s + 1.22·6-s − 1.13·7-s − 0.707·8-s + 2·9-s + 0.632·10-s − 3.01·11-s − 1.73·12-s − 0.801·14-s + 1.54·15-s + 1/4·16-s + 1.41·18-s − 1.83·19-s − 0.894·20-s − 1.96·21-s − 2.13·22-s − 0.417·23-s − 1.22·24-s − 3/5·25-s + 1.92·27-s + 1.13·28-s − 0.371·29-s + 1.09·30-s − 2.15·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + 3 T^{2} - 3 T^{3} + 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 24 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 10 T + 61 T^{2} + 240 T^{3} + 61 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 272 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 49 T^{2} + 84 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 12 T + 77 T^{2} + 424 T^{3} + 77 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 95 T^{2} + 264 T^{3} + 95 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 14 T + 155 T^{2} + 1168 T^{3} + 155 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 16 T + 161 T^{2} - 1248 T^{3} + 161 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 113 T^{2} - 52 T^{3} + 113 p T^{4} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 189 T^{2} + 948 T^{3} + 189 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 11 T^{2} + 684 T^{3} + 11 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 16 T + 265 T^{2} + 2176 T^{3} + 265 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 65 T^{2} - 112 T^{3} + 65 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 16 T + 211 T^{2} + 2128 T^{3} + 211 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 309 T^{2} + 2608 T^{3} + 309 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 77 T^{2} + 76 T^{3} + 77 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 179 T^{2} + 1144 T^{3} + 179 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 275 T^{2} + 16 T^{3} + 275 p T^{4} + p^{3} T^{6} \) |
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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
−8.026470403754430928301888512930, −7.66969512564356550043956271443, −7.51221956300451899195394864370, −7.41064435114812906123342812824, −6.96040202029186058894948315348, −6.83880109159493490394988219385, −6.43138722427240160447515634939, −5.97508065388799557487457795239, −5.86064063314175302117885128179, −5.76848489580503847188756132254, −5.31863780019766886909067145680, −5.04330297313452013854579138019, −5.01672421669235730973178886135, −4.40973790329349313583765599093, −4.25290270243069877042342930923, −4.20280950530329217325703853481, −3.70813976527489658258187279913, −3.41927488060824146090690480056, −3.13849694648299946041748985508, −2.95977847728260833218594699578, −2.54905936482338173815289155134, −2.28438873443305269883375726399, −2.07959567422964930602298374869, −1.69569860922406543969225456123, −1.34517993681903708418920664535, 0, 0, 0,
1.34517993681903708418920664535, 1.69569860922406543969225456123, 2.07959567422964930602298374869, 2.28438873443305269883375726399, 2.54905936482338173815289155134, 2.95977847728260833218594699578, 3.13849694648299946041748985508, 3.41927488060824146090690480056, 3.70813976527489658258187279913, 4.20280950530329217325703853481, 4.25290270243069877042342930923, 4.40973790329349313583765599093, 5.01672421669235730973178886135, 5.04330297313452013854579138019, 5.31863780019766886909067145680, 5.76848489580503847188756132254, 5.86064063314175302117885128179, 5.97508065388799557487457795239, 6.43138722427240160447515634939, 6.83880109159493490394988219385, 6.96040202029186058894948315348, 7.41064435114812906123342812824, 7.51221956300451899195394864370, 7.66969512564356550043956271443, 8.026470403754430928301888512930
