| L(s) = 1 | − 3-s − 3·5-s − 3·7-s − 9-s − 2·13-s + 3·15-s + 4·17-s + 4·19-s + 3·21-s + 6·23-s + 6·25-s − 6·27-s + 2·29-s + 22·31-s + 9·35-s − 4·37-s + 2·39-s + 5·41-s + 43-s + 3·45-s + 7·47-s + 5·49-s − 4·51-s − 6·53-s − 4·57-s + 6·59-s − 21·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 1.13·7-s − 1/3·9-s − 0.554·13-s + 0.774·15-s + 0.970·17-s + 0.917·19-s + 0.654·21-s + 1.25·23-s + 6/5·25-s − 1.15·27-s + 0.371·29-s + 3.95·31-s + 1.52·35-s − 0.657·37-s + 0.320·39-s + 0.780·41-s + 0.152·43-s + 0.447·45-s + 1.02·47-s + 5/7·49-s − 0.560·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s − 2.68·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7316596388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7316596388\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 2 T^{2} + p^{2} T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 4 T^{2} - 11 T^{3} + 4 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 31 T^{2} + 48 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $D_{6}$ | \( 1 - 4 T + 3 T^{2} + 8 T^{3} + 3 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 140 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 49 T^{2} - 264 T^{3} + 49 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 59 T^{2} - 140 T^{3} + 59 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 22 T + 245 T^{2} - 1688 T^{3} + 245 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 87 T^{2} + 216 T^{3} + 87 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 5 T + 122 T^{2} - 401 T^{3} + 122 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - T + 48 T^{2} + 189 T^{3} + 48 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 7 T + 106 T^{2} - 667 T^{3} + 106 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 8 p T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 157 T^{2} - 696 T^{3} + 157 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 21 T + 298 T^{2} + 2637 T^{3} + 298 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 15 T + 210 T^{2} + 1659 T^{3} + 210 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 10 T + 17 T^{2} + 576 T^{3} + 17 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 187 T^{2} + 552 T^{3} + 187 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 185 T^{2} + 356 T^{3} + 185 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 237 T^{2} + 1480 T^{3} + 237 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 19 T + 334 T^{2} - 3223 T^{3} + 334 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 283 T^{2} + 384 T^{3} + 283 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.74349977448264297050219478326, −6.39256081955970848732022166314, −6.33400436189944500004257764088, −6.33287136106422449966741745012, −5.81828769912611890764890378194, −5.63168237362235861197257656547, −5.49861301779347444702456909331, −5.10802287445938806358510148538, −4.85880041861123972144030740543, −4.75819864839512355821691914773, −4.38140052411587013889669215195, −4.22145191342851592529811151276, −4.08498314296524099218806693668, −3.50903433484823601435748132993, −3.38987051750791996852765831353, −3.23268152046293798263464044449, −2.80077240420415445198422013430, −2.77500791584954561495964684711, −2.67442577259678371086511139167, −1.93992602894495826879569215722, −1.74763760220398179385037924492, −1.03163614603251406388113488276, −0.967691583183578732791933198747, −0.70159063068995238089544857894, −0.18633535267026434579570679764,
0.18633535267026434579570679764, 0.70159063068995238089544857894, 0.967691583183578732791933198747, 1.03163614603251406388113488276, 1.74763760220398179385037924492, 1.93992602894495826879569215722, 2.67442577259678371086511139167, 2.77500791584954561495964684711, 2.80077240420415445198422013430, 3.23268152046293798263464044449, 3.38987051750791996852765831353, 3.50903433484823601435748132993, 4.08498314296524099218806693668, 4.22145191342851592529811151276, 4.38140052411587013889669215195, 4.75819864839512355821691914773, 4.85880041861123972144030740543, 5.10802287445938806358510148538, 5.49861301779347444702456909331, 5.63168237362235861197257656547, 5.81828769912611890764890378194, 6.33287136106422449966741745012, 6.33400436189944500004257764088, 6.39256081955970848732022166314, 6.74349977448264297050219478326
