| L(s) = 1 | + 4·3-s + 3·5-s − 3·7-s + 5·9-s − 3·11-s − 4·13-s + 12·15-s − 6·17-s + 6·19-s − 12·21-s + 6·23-s + 6·25-s − 2·27-s − 2·29-s + 16·31-s − 12·33-s − 9·35-s − 16·39-s + 2·41-s − 2·43-s + 15·45-s + 14·47-s + 6·49-s − 24·51-s + 4·53-s − 9·55-s + 24·57-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.34·5-s − 1.13·7-s + 5/3·9-s − 0.904·11-s − 1.10·13-s + 3.09·15-s − 1.45·17-s + 1.37·19-s − 2.61·21-s + 1.25·23-s + 6/5·25-s − 0.384·27-s − 0.371·29-s + 2.87·31-s − 2.08·33-s − 1.52·35-s − 2.56·39-s + 0.312·41-s − 0.304·43-s + 2.23·45-s + 2.04·47-s + 6/7·49-s − 3.36·51-s + 0.549·53-s − 1.21·55-s + 3.17·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.51180189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.51180189\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 4 T + 11 T^{2} - 22 T^{3} + 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 29 T^{2} + 66 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 29 T^{2} + 82 T^{3} + 29 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T - 3 T^{2} + 152 T^{3} - 3 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 67 T^{2} + 108 T^{3} + 67 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 16 T + 165 T^{2} - 34 p T^{3} + 165 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 83 T^{2} - 52 T^{3} + 83 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 111 T^{2} - 174 T^{3} + 111 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 85 T^{2} + 152 T^{3} + 85 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 14 T + 131 T^{2} - 1042 T^{3} + 131 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 4 T + 51 T^{2} - 372 T^{3} + 51 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 149 T^{2} - 730 T^{3} + 149 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 4 T + 175 T^{2} + 454 T^{3} + 175 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 189 T^{2} - 1216 T^{3} + 189 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 53 T^{2} - 608 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 173 T^{2} + 598 T^{3} + 173 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 101 T^{2} + 4 p T^{3} + 101 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 22 T + 325 T^{2} - 3468 T^{3} + 325 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 20 T + 315 T^{2} - 3240 T^{3} + 315 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 275 T^{2} + 744 T^{3} + 275 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.30408405928309870132225363970, −7.05616978446534977570166668575, −6.59414757620783330404479338062, −6.52894859526747951796464101108, −6.10172651411149458639832499984, −5.93966178778670592123176202285, −5.91532516324558604522887449136, −5.26522547180109285388601786852, −5.14996184054903739870265851900, −5.08197493311618451942983977182, −4.59377291086146991949441645635, −4.39377445124551597692025218111, −4.26675212733026237765606165712, −3.53767611109991135476879695970, −3.45666542145030456517968976063, −3.29568564077501148721082169067, −2.85244257106196739261030554700, −2.72094473034636965636700398306, −2.69172241372341170808880270331, −2.13345634070487163539284745946, −2.11758107576544384567453247420, −2.00536045342019341844208949719, −1.06893896768011157585719842423, −0.78537555027959166983619665562, −0.52660627755566654147193779716,
0.52660627755566654147193779716, 0.78537555027959166983619665562, 1.06893896768011157585719842423, 2.00536045342019341844208949719, 2.11758107576544384567453247420, 2.13345634070487163539284745946, 2.69172241372341170808880270331, 2.72094473034636965636700398306, 2.85244257106196739261030554700, 3.29568564077501148721082169067, 3.45666542145030456517968976063, 3.53767611109991135476879695970, 4.26675212733026237765606165712, 4.39377445124551597692025218111, 4.59377291086146991949441645635, 5.08197493311618451942983977182, 5.14996184054903739870265851900, 5.26522547180109285388601786852, 5.91532516324558604522887449136, 5.93966178778670592123176202285, 6.10172651411149458639832499984, 6.52894859526747951796464101108, 6.59414757620783330404479338062, 7.05616978446534977570166668575, 7.30408405928309870132225363970
