| L(s) = 1 | − 2·3-s + 3·5-s + 3·7-s + 9-s − 3·11-s + 2·13-s − 6·15-s + 2·17-s + 6·19-s − 6·21-s − 10·23-s + 6·25-s + 12·31-s + 6·33-s + 9·35-s + 4·37-s − 4·39-s + 4·41-s − 8·43-s + 3·45-s + 6·49-s − 4·51-s + 12·53-s − 9·55-s − 12·57-s + 4·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.554·13-s − 1.54·15-s + 0.485·17-s + 1.37·19-s − 1.30·21-s − 2.08·23-s + 6/5·25-s + 2.15·31-s + 1.04·33-s + 1.52·35-s + 0.657·37-s − 0.640·39-s + 0.624·41-s − 1.21·43-s + 0.447·45-s + 6/7·49-s − 0.560·51-s + 1.64·53-s − 1.21·55-s − 1.58·57-s + 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.463480510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.463480510\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 23 T^{2} - 36 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 23 T^{2} - 76 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 11 T^{2} + 4 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 10 T + 95 T^{2} + 476 T^{3} + 95 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 92 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 12 T + 113 T^{2} - 712 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 109 T^{2} - 292 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} - 12 p T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 21 T^{2} - 160 T^{3} + 21 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 29 T^{2} + 128 T^{3} + 29 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 12 T + 141 T^{2} - 980 T^{3} + 141 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 113 T^{2} - 344 T^{3} + 113 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 83 T^{2} + 388 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 89 T^{2} + 600 T^{3} + 89 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 28 T + 413 T^{2} - 4040 T^{3} + 413 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 14 T + 255 T^{2} + 2036 T^{3} + 255 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 18 T + 287 T^{2} + 2588 T^{3} + 287 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 241 T^{2} - 1296 T^{3} + 241 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 239 T^{2} - 1996 T^{3} + 239 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 137 T^{2} + 1092 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.26765949487601721034436396335, −6.69907531909780803573580330650, −6.54955471571434654483315538366, −6.38811394097248376949008172520, −5.94028866958454666283743909286, −5.93841329787691741526390475697, −5.77760234751686823992690139057, −5.40420467309929948111726515515, −5.24706741901246797976673543355, −5.14887473730234402834845346183, −4.79629153123379106836435646031, −4.46695675645097904113592165854, −4.36009587098081059962427771667, −3.92373527899100765570343554066, −3.76825815807959980425604170294, −3.28820645138040150853821611501, −2.94534039305637134397926774716, −2.64505691466926336890008562081, −2.59884627036991048519611456286, −1.94506602457897269567454038108, −1.82423206261066086749549891403, −1.59951909010789060683837956891, −1.06477392838051619790040730069, −0.65214126911301154548990563558, −0.56185200369579379958390269150,
0.56185200369579379958390269150, 0.65214126911301154548990563558, 1.06477392838051619790040730069, 1.59951909010789060683837956891, 1.82423206261066086749549891403, 1.94506602457897269567454038108, 2.59884627036991048519611456286, 2.64505691466926336890008562081, 2.94534039305637134397926774716, 3.28820645138040150853821611501, 3.76825815807959980425604170294, 3.92373527899100765570343554066, 4.36009587098081059962427771667, 4.46695675645097904113592165854, 4.79629153123379106836435646031, 5.14887473730234402834845346183, 5.24706741901246797976673543355, 5.40420467309929948111726515515, 5.77760234751686823992690139057, 5.93841329787691741526390475697, 5.94028866958454666283743909286, 6.38811394097248376949008172520, 6.54955471571434654483315538366, 6.69907531909780803573580330650, 7.26765949487601721034436396335
