| L(s) = 1 | − 3·2-s + 2·3-s + 6·4-s + 2·5-s − 6·6-s + 4·7-s − 10·8-s − 3·9-s − 6·10-s − 3·11-s + 12·12-s + 7·13-s − 12·14-s + 4·15-s + 15·16-s + 16·17-s + 9·18-s + 12·20-s + 8·21-s + 9·22-s + 8·23-s − 20·24-s + 25-s − 21·26-s − 10·27-s + 24·28-s + 5·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.15·3-s + 3·4-s + 0.894·5-s − 2.44·6-s + 1.51·7-s − 3.53·8-s − 9-s − 1.89·10-s − 0.904·11-s + 3.46·12-s + 1.94·13-s − 3.20·14-s + 1.03·15-s + 15/4·16-s + 3.88·17-s + 2.12·18-s + 2.68·20-s + 1.74·21-s + 1.91·22-s + 1.66·23-s − 4.08·24-s + 1/5·25-s − 4.11·26-s − 1.92·27-s + 4.53·28-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.624493909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.624493909\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 10 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $D_{6}$ | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 11 T^{2} - 18 T^{3} + 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 7 T + 46 T^{2} - 163 T^{3} + 46 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 16 T + 123 T^{2} - 606 T^{3} + 123 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 29 T^{2} - 64 T^{3} + 29 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 74 T^{2} - 265 T^{3} + 74 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 376 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 91 T^{2} - 98 T^{3} + 91 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 119 T^{2} + 2 T^{3} + 119 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - T + 116 T^{2} - 63 T^{3} + 116 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 134 T^{2} + 833 T^{3} + 134 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 4 T + 151 T^{2} - 408 T^{3} + 151 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 161 T^{2} + 16 T^{3} + 161 p T^{4} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 21 T + 282 T^{2} - 2549 T^{3} + 282 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 14 T + 191 T^{2} - 1602 T^{3} + 191 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 3 T + 116 T^{2} - 577 T^{3} + 116 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 26 T + 395 T^{2} + 3918 T^{3} + 395 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 20 T + 199 T^{2} + 1502 T^{3} + 199 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 7 T + 172 T^{2} - 1293 T^{3} + 172 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 23 T + 350 T^{2} - 3691 T^{3} + 350 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 13 T - 26 T^{2} - 1703 T^{3} - 26 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.15652631429920029559927054262, −7.00496002468883466086781489556, −6.41701404414112859390672282494, −6.31939636267114803608115050464, −5.93445735797865832229806105467, −5.74098666701294734341752541758, −5.70213608956936243470430208894, −5.43374860019921474170566070997, −5.20448099828401928901955391210, −4.89781079582474580488355550770, −4.78848174756802841915485965878, −4.06949378326965554472641467693, −3.85861822897935330157937225408, −3.43511178664391565467328872129, −3.30455060667908499379558807887, −3.18940679325509908098507596579, −2.77634412136054651170548659686, −2.67752753072962604255970992459, −2.37177497775907111290550933325, −1.74612259927829531938363208142, −1.74566076339013601753578101498, −1.51402409256826151896774147523, −1.00763726141246579366058064627, −0.811168925429281440388766300748, −0.57892534196560221117922783480,
0.57892534196560221117922783480, 0.811168925429281440388766300748, 1.00763726141246579366058064627, 1.51402409256826151896774147523, 1.74566076339013601753578101498, 1.74612259927829531938363208142, 2.37177497775907111290550933325, 2.67752753072962604255970992459, 2.77634412136054651170548659686, 3.18940679325509908098507596579, 3.30455060667908499379558807887, 3.43511178664391565467328872129, 3.85861822897935330157937225408, 4.06949378326965554472641467693, 4.78848174756802841915485965878, 4.89781079582474580488355550770, 5.20448099828401928901955391210, 5.43374860019921474170566070997, 5.70213608956936243470430208894, 5.74098666701294734341752541758, 5.93445735797865832229806105467, 6.31939636267114803608115050464, 6.41701404414112859390672282494, 7.00496002468883466086781489556, 7.15652631429920029559927054262
