| L(s) = 1 | + 3-s − 5·5-s − 3·7-s − 9-s − 3·11-s + 4·13-s − 5·15-s + 6·17-s + 3·19-s − 3·21-s − 2·23-s + 7·25-s + 7·27-s − 5·29-s + 4·31-s − 3·33-s + 15·35-s − 7·37-s + 4·39-s − 7·41-s + 43-s + 5·45-s − 11·47-s + 6·49-s + 6·51-s − 3·53-s + 15·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.23·5-s − 1.13·7-s − 1/3·9-s − 0.904·11-s + 1.10·13-s − 1.29·15-s + 1.45·17-s + 0.688·19-s − 0.654·21-s − 0.417·23-s + 7/5·25-s + 1.34·27-s − 0.928·29-s + 0.718·31-s − 0.522·33-s + 2.53·35-s − 1.15·37-s + 0.640·39-s − 1.09·41-s + 0.152·43-s + 0.745·45-s − 1.60·47-s + 6/7·49-s + 0.840·51-s − 0.412·53-s + 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4968647924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4968647924\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{2} - 10 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + p T + 18 T^{2} + 48 T^{3} + 18 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 8 T^{2} - 10 T^{3} + 8 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 23 T^{2} - 96 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 11 T^{2} + 20 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 49 T^{2} + 60 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 60 T^{2} + 252 T^{3} + 60 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 184 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 7 T + 92 T^{2} + 432 T^{3} + 92 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 7 T + 134 T^{2} + 572 T^{3} + 134 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - T + 94 T^{2} - 58 T^{3} + 94 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 11 T + 174 T^{2} + 1050 T^{3} + 174 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 3 T + 96 T^{2} + 80 T^{3} + 96 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 3 T + 132 T^{2} - 246 T^{3} + 132 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 7 T + 170 T^{2} + 852 T^{3} + 170 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 197 T^{2} + 1592 T^{3} + 197 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 9 T + 212 T^{2} - 1270 T^{3} + 212 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 107 T^{2} - 392 T^{3} + 107 p T^{4} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 15 T + 278 T^{2} - 2386 T^{3} + 278 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 2208 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 242 T^{2} + 556 T^{3} + 242 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 5 T + 270 T^{2} + 872 T^{3} + 270 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.99134866970205410584073472677, −6.72347206740330756922509527471, −6.49017833760242067682406754301, −6.24045327514215563466098434889, −5.81560647504362755593632513161, −5.78983308678956390963276266188, −5.63164392363462051167165244564, −5.03384254177756905204361168002, −4.97050538025636767406596639677, −4.79880106555518917897537881696, −4.52081430612684984017519446993, −4.01090025700877096762517552198, −3.95764933623508323122314791097, −3.49642310962435476659308171479, −3.49242284823711334910908612793, −3.30657126768425162594755543469, −3.17863724697596094662929168042, −2.83928690627076691724276179248, −2.54337538275246779834520903681, −2.05298566418644520473514750864, −1.81199737199354855324523860259, −1.36698627201529995067622304909, −0.896317434534893010150057639540, −0.62163883261231003078288313739, −0.15321429453026988786426991553,
0.15321429453026988786426991553, 0.62163883261231003078288313739, 0.896317434534893010150057639540, 1.36698627201529995067622304909, 1.81199737199354855324523860259, 2.05298566418644520473514750864, 2.54337538275246779834520903681, 2.83928690627076691724276179248, 3.17863724697596094662929168042, 3.30657126768425162594755543469, 3.49242284823711334910908612793, 3.49642310962435476659308171479, 3.95764933623508323122314791097, 4.01090025700877096762517552198, 4.52081430612684984017519446993, 4.79880106555518917897537881696, 4.97050538025636767406596639677, 5.03384254177756905204361168002, 5.63164392363462051167165244564, 5.78983308678956390963276266188, 5.81560647504362755593632513161, 6.24045327514215563466098434889, 6.49017833760242067682406754301, 6.72347206740330756922509527471, 6.99134866970205410584073472677
