| L(s) = 1 | + 3·2-s + 6·4-s + 5-s + 10·8-s + 3·10-s − 11-s − 8·13-s + 15·16-s − 4·17-s + 3·19-s + 6·20-s − 3·22-s − 7·23-s − 8·25-s − 24·26-s − 5·29-s − 20·31-s + 21·32-s − 12·34-s − 3·37-s + 9·38-s + 10·40-s + 6·43-s − 6·44-s − 21·46-s − 9·47-s − 24·50-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 0.447·5-s + 3.53·8-s + 0.948·10-s − 0.301·11-s − 2.21·13-s + 15/4·16-s − 0.970·17-s + 0.688·19-s + 1.34·20-s − 0.639·22-s − 1.45·23-s − 8/5·25-s − 4.70·26-s − 0.928·29-s − 3.59·31-s + 3.71·32-s − 2.05·34-s − 0.493·37-s + 1.45·38-s + 1.58·40-s + 0.914·43-s − 0.904·44-s − 3.09·46-s − 1.31·47-s − 3.39·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 - T + 9 T^{2} - 13 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + T + 27 T^{2} + 25 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 40 T^{2} + 139 T^{3} + 40 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 39 T^{2} + 112 T^{3} + 39 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 81 T^{2} + 325 T^{3} + 81 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 21 T^{2} - 73 T^{3} + 21 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 20 T + 214 T^{2} + 1441 T^{3} + 214 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 90 T^{2} - 9 T^{3} + 90 p T^{4} + p^{3} T^{6} \) |
| 43 | $D_{6}$ | \( 1 - 6 T + 60 T^{2} - 389 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 87 T^{2} + 657 T^{3} + 87 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 15 T + 225 T^{2} - 1671 T^{3} + 225 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 216 T^{2} + 1589 T^{3} + 216 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 178 T^{2} + 883 T^{3} + 178 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + 89 T^{2} - 77 T^{3} + 89 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 7 T + 15 T^{2} - 599 T^{3} + 15 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 19 T + 227 T^{2} + 2143 T^{3} + 227 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 5 T + 163 T^{2} + 469 T^{3} + 163 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 186 T^{2} - 185 T^{3} + 186 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 9 T + 225 T^{2} + 1611 T^{3} + 225 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 28 T + 503 T^{2} + 5680 T^{3} + 503 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.23320282968054898379979169110, −6.85688085320084960690077356140, −6.85134296528715666825744301319, −6.68864714091339627681118938886, −6.08729244675036512384235161271, −5.91114634036461118541080250933, −5.78056478819806350629882335070, −5.59773769679109872781060547091, −5.42952197015944409339842031911, −5.26087259370471126910976958726, −4.95394597661199065874374072953, −4.55372119486121625516737077535, −4.50150347124817876303999759132, −4.09874990536538524243320413195, −3.95665007602138802688391405420, −3.89293507264056159849208223328, −3.39207575560048019425319927765, −3.10588282557686314366569160270, −2.96511428773177502628389546963, −2.51297238425872211529891621445, −2.23903296957639121881668224279, −2.22383950695198414951966717987, −1.76578951812068376762443585300, −1.52204451592684212604452272538, −1.36298111872214932199938289561, 0, 0, 0,
1.36298111872214932199938289561, 1.52204451592684212604452272538, 1.76578951812068376762443585300, 2.22383950695198414951966717987, 2.23903296957639121881668224279, 2.51297238425872211529891621445, 2.96511428773177502628389546963, 3.10588282557686314366569160270, 3.39207575560048019425319927765, 3.89293507264056159849208223328, 3.95665007602138802688391405420, 4.09874990536538524243320413195, 4.50150347124817876303999759132, 4.55372119486121625516737077535, 4.95394597661199065874374072953, 5.26087259370471126910976958726, 5.42952197015944409339842031911, 5.59773769679109872781060547091, 5.78056478819806350629882335070, 5.91114634036461118541080250933, 6.08729244675036512384235161271, 6.68864714091339627681118938886, 6.85134296528715666825744301319, 6.85688085320084960690077356140, 7.23320282968054898379979169110
