| L(s) = 1 | − 3·2-s + 6·4-s − 3·5-s − 3·7-s − 10·8-s + 9-s + 9·10-s + 3·11-s + 9·14-s + 15·16-s − 8·17-s − 3·18-s − 2·19-s − 18·20-s − 9·22-s + 2·23-s + 6·25-s + 4·27-s − 18·28-s + 4·29-s + 18·31-s − 21·32-s + 24·34-s + 9·35-s + 6·36-s + 4·37-s + 6·38-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 1.34·5-s − 1.13·7-s − 3.53·8-s + 1/3·9-s + 2.84·10-s + 0.904·11-s + 2.40·14-s + 15/4·16-s − 1.94·17-s − 0.707·18-s − 0.458·19-s − 4.02·20-s − 1.91·22-s + 0.417·23-s + 6/5·25-s + 0.769·27-s − 3.40·28-s + 0.742·29-s + 3.23·31-s − 3.71·32-s + 4.11·34-s + 1.52·35-s + 36-s + 0.657·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5173814688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5173814688\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T^{2} - 4 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - T^{2} - 32 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 39 T^{2} + 144 T^{3} + 39 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 27 T^{2} + 20 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 39 T^{2} - 36 T^{3} + 39 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 196 T^{3} + 5 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 172 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 18 T + 221 T^{2} + 1636 T^{3} + 221 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 101 T^{2} - 672 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 93 T^{2} - 252 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 4 T + 77 T^{2} + 4 T^{3} + 77 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 10 T + 113 T^{2} - 572 T^{3} + 113 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 20 T + 267 T^{2} - 2504 T^{3} + 267 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 109 T^{2} - 1104 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 79 T^{2} - 200 T^{3} + 79 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T - 25 T^{2} - 1324 T^{3} - 25 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 24 T + 401 T^{2} + 4144 T^{3} + 401 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 239 T^{2} + 1028 T^{3} + 239 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 281 T^{2} - 1452 T^{3} + 281 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.082367021117925585255363920823, −8.761240085423203053334663276038, −8.546578666809495227007706861792, −8.437358141955009494890152436434, −8.290384835808360788561045695506, −7.988769677819698182606746893012, −7.40656788026391576769982919287, −6.95938331858107350714800458445, −6.89532147973069709931972235948, −6.88920891639272717800595420490, −6.52701242703705687737240559725, −6.08454041736139468132213859871, −6.05690711894951858180856458978, −5.08301946835727542382005756075, −5.02242371928895714779292390909, −4.40760981648315973252342226898, −4.16103565729045691634104870649, −3.62616469844236481638353617141, −3.58299491889442570350027243061, −2.66362652996341208976806759276, −2.64879365254247449832644212716, −2.32463326999289407275773092013, −1.43465852026075341538500266859, −0.828623097293271052336299898781, −0.52551632036519081960594518074,
0.52551632036519081960594518074, 0.828623097293271052336299898781, 1.43465852026075341538500266859, 2.32463326999289407275773092013, 2.64879365254247449832644212716, 2.66362652996341208976806759276, 3.58299491889442570350027243061, 3.62616469844236481638353617141, 4.16103565729045691634104870649, 4.40760981648315973252342226898, 5.02242371928895714779292390909, 5.08301946835727542382005756075, 6.05690711894951858180856458978, 6.08454041736139468132213859871, 6.52701242703705687737240559725, 6.88920891639272717800595420490, 6.89532147973069709931972235948, 6.95938331858107350714800458445, 7.40656788026391576769982919287, 7.988769677819698182606746893012, 8.290384835808360788561045695506, 8.437358141955009494890152436434, 8.546578666809495227007706861792, 8.761240085423203053334663276038, 9.082367021117925585255363920823
