| L(s) = 1 | + 3·2-s + 3-s + 6·4-s + 5·5-s + 3·6-s + 7-s + 10·8-s − 3·9-s + 15·10-s − 3·11-s + 6·12-s + 5·13-s + 3·14-s + 5·15-s + 15·16-s + 4·17-s − 9·18-s − 3·19-s + 30·20-s + 21-s − 9·22-s + 2·23-s + 10·24-s + 7·25-s + 15·26-s − 5·27-s + 6·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 3·4-s + 2.23·5-s + 1.22·6-s + 0.377·7-s + 3.53·8-s − 9-s + 4.74·10-s − 0.904·11-s + 1.73·12-s + 1.38·13-s + 0.801·14-s + 1.29·15-s + 15/4·16-s + 0.970·17-s − 2.12·18-s − 0.688·19-s + 6.70·20-s + 0.218·21-s − 1.91·22-s + 0.417·23-s + 2.04·24-s + 7/5·25-s + 2.94·26-s − 0.962·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73034632 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73034632 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.21455613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.21455613\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 4 T^{2} - 2 T^{3} + 4 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - p T + 18 T^{2} - 48 T^{3} + 18 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 18 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 40 T^{2} - 116 T^{3} + 40 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 27 T^{2} - 48 T^{3} + 27 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 41 T^{2} - 124 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 68 T^{2} - 320 T^{3} + 68 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 68 T^{2} + 110 T^{3} + 68 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 14 T + 127 T^{2} + 908 T^{3} + 127 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 3 T + 98 T^{2} + 268 T^{3} + 98 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 5 T + 62 T^{2} + 162 T^{3} + 62 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 89 T^{2} + 128 T^{3} + 89 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 223 T^{2} + 1752 T^{3} + 223 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 17 T^{2} - 376 T^{3} + 17 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - T + 194 T^{2} - 138 T^{3} + 194 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 3 T + 182 T^{2} + 454 T^{3} + 182 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 4 T + 83 T^{2} - 528 T^{3} + 83 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 121 T^{2} - 668 T^{3} + 121 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 7 T + 106 T^{2} - 54 T^{3} + 106 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16 T - 13 T^{2} + 1576 T^{3} - 13 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 307 T^{2} + 2588 T^{3} + 307 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.16050325440825332556907548324, −9.828055502274430804721902238363, −9.400503517462249451900216195901, −9.250074976273593621794054823911, −8.540511508341507227250707888885, −8.432071805849707563616367264204, −8.275157492226730483224248510818, −7.69558298026064181626568328514, −7.52081400967267636097075807942, −6.86228195712903422927850873640, −6.46899371741935064110887626858, −6.33486490588264623934865299008, −6.12474216890903363498684479580, −5.61935710735571675983611659647, −5.41305306052668382208684587190, −5.25052143021557226933800496283, −4.78676607391630482467669756030, −4.50628536033226800674415192783, −3.75176066846497908511832953172, −3.38782446174387078263704721746, −3.01107070329188969587230374623, −2.93214087583392821062183713820, −1.99354725019068991302511362310, −1.92206372717885493143257013078, −1.50771487045926067104229885805,
1.50771487045926067104229885805, 1.92206372717885493143257013078, 1.99354725019068991302511362310, 2.93214087583392821062183713820, 3.01107070329188969587230374623, 3.38782446174387078263704721746, 3.75176066846497908511832953172, 4.50628536033226800674415192783, 4.78676607391630482467669756030, 5.25052143021557226933800496283, 5.41305306052668382208684587190, 5.61935710735571675983611659647, 6.12474216890903363498684479580, 6.33486490588264623934865299008, 6.46899371741935064110887626858, 6.86228195712903422927850873640, 7.52081400967267636097075807942, 7.69558298026064181626568328514, 8.275157492226730483224248510818, 8.432071805849707563616367264204, 8.540511508341507227250707888885, 9.250074976273593621794054823911, 9.400503517462249451900216195901, 9.828055502274430804721902238363, 10.16050325440825332556907548324
