| L(s) = 1 | − 3·2-s + 4·3-s + 4·4-s + 3·5-s − 12·6-s − 4·8-s + 5·9-s − 9·10-s + 3·11-s + 16·12-s + 8·13-s + 12·15-s + 3·16-s + 14·17-s − 15·18-s − 2·19-s + 12·20-s − 9·22-s − 6·23-s − 16·24-s + 6·25-s − 24·26-s − 2·27-s + 2·29-s − 36·30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2.30·3-s + 2·4-s + 1.34·5-s − 4.89·6-s − 1.41·8-s + 5/3·9-s − 2.84·10-s + 0.904·11-s + 4.61·12-s + 2.21·13-s + 3.09·15-s + 3/4·16-s + 3.39·17-s − 3.53·18-s − 0.458·19-s + 2.68·20-s − 1.91·22-s − 1.25·23-s − 3.26·24-s + 6/5·25-s − 4.70·26-s − 0.384·27-s + 0.371·29-s − 6.57·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.900695356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.900695356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 3 T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 - 8 T + 57 T^{2} - 218 T^{3} + 57 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 14 T + 113 T^{2} - 562 T^{3} + 113 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 124 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 128 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 51 T^{2} - 12 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} - 298 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 35 T^{2} - 156 T^{3} + 35 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 135 T^{2} - 822 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 - 26 T + 363 T^{2} - 3066 T^{3} + 363 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 4 T + 67 T^{2} - 492 T^{3} + 67 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 161 T^{2} - 814 T^{3} + 161 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 191 T^{2} - 978 T^{3} + 191 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 113 T^{2} + 880 T^{3} + 113 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 20 T + 325 T^{2} + 3000 T^{3} + 325 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 161 T^{2} - 830 T^{3} + 161 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 8 T + 125 T^{2} + 1020 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 101 T^{2} + 1172 T^{3} + 101 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16 T + 331 T^{2} - 2880 T^{3} + 331 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 227 T^{2} + 128 T^{3} + 227 p T^{4} + p^{3} T^{6} \) |
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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
−8.067360394815936032060835518047, −7.85053718147858276801687321843, −7.46540798749994557457233480184, −7.45999426689692318694473780558, −6.90008494906064655852947412926, −6.79854620507438846597463613866, −6.13698867291239538994382008555, −6.06238664709498998771517137722, −5.89408530772367878339645385696, −5.75292630469813870874575019884, −5.53331620182192644984551085573, −4.96490222140628190201365359973, −4.46777784772091245631287197271, −4.05599675233921945551006723048, −3.83314358220441212368194207507, −3.67692223752058728304627945295, −3.33110215329385619342542164004, −2.78973761766390815252425111673, −2.76061541662585513888646306573, −2.50075049751627098615027948491, −2.06714818013224400054622978707, −1.66266634919272501604327845413, −1.09909048434647778246207616357, −1.05955041249915747520289383113, −0.74842371782365523476453750851,
0.74842371782365523476453750851, 1.05955041249915747520289383113, 1.09909048434647778246207616357, 1.66266634919272501604327845413, 2.06714818013224400054622978707, 2.50075049751627098615027948491, 2.76061541662585513888646306573, 2.78973761766390815252425111673, 3.33110215329385619342542164004, 3.67692223752058728304627945295, 3.83314358220441212368194207507, 4.05599675233921945551006723048, 4.46777784772091245631287197271, 4.96490222140628190201365359973, 5.53331620182192644984551085573, 5.75292630469813870874575019884, 5.89408530772367878339645385696, 6.06238664709498998771517137722, 6.13698867291239538994382008555, 6.79854620507438846597463613866, 6.90008494906064655852947412926, 7.45999426689692318694473780558, 7.46540798749994557457233480184, 7.85053718147858276801687321843, 8.067360394815936032060835518047
