| L(s) = 1 | − 3-s − 5-s + 3·7-s − 2·9-s + 3·11-s + 12·13-s + 15-s + 2·17-s − 2·19-s − 3·21-s − 7·23-s + 2·25-s + 27-s + 6·29-s − 9·31-s − 3·33-s − 3·35-s + 7·37-s − 12·39-s − 2·41-s − 4·43-s + 2·45-s − 4·47-s + 6·49-s − 2·51-s + 2·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s + 0.904·11-s + 3.32·13-s + 0.258·15-s + 0.485·17-s − 0.458·19-s − 0.654·21-s − 1.45·23-s + 2/5·25-s + 0.192·27-s + 1.11·29-s − 1.61·31-s − 0.522·33-s − 0.507·35-s + 1.15·37-s − 1.92·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s − 0.583·47-s + 6/7·49-s − 0.280·51-s + 0.274·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29218112 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29218112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.677897124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677897124\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 4 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + T - T^{2} - 2 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 304 T^{3} + 73 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} + 88 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 33 T^{2} + 60 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 61 T^{2} + 250 T^{3} + 61 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 43 T^{2} - 372 T^{3} + 43 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 9 T + 63 T^{2} + 412 T^{3} + 63 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 7 T + 3 p T^{2} - 486 T^{3} + 3 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 77 T^{2} + 8 T^{3} + 77 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 109 T^{2} + 312 T^{3} + 109 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 115 T^{2} + 280 T^{3} + 115 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 95 T^{2} - 116 T^{3} + 95 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 23 T + 347 T^{2} + 3116 T^{3} + 347 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 121 T^{2} - 792 T^{3} + 121 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 5 T + 185 T^{2} + 662 T^{3} + 185 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 5 T + 101 T^{2} + 1022 T^{3} + 101 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 14 T + 237 T^{2} + 1800 T^{3} + 237 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 177 T^{2} - 1276 T^{3} + 177 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 65 T^{2} + 584 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 19 T + 235 T^{2} + 2098 T^{3} + 235 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 15 T + 307 T^{2} - 2914 T^{3} + 307 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−10.55146886796298924031160393097, −10.36104760247380131767828863417, −9.937723593079180218837299839198, −9.421287975559228696494609430797, −8.974582059642785497132831744137, −8.846778275545068922937410317921, −8.600601626661742321236689991295, −8.081731312542757679368832380004, −8.017718260200687266378339625891, −7.80832452503619593074899096088, −7.08878816501079040308285094333, −6.71065097447877039768939551027, −6.46320083257403540216448331569, −5.82414629923919809239949169271, −5.79522770298325749593257454279, −5.78547542369211224423224850811, −4.89996582576554091537221458296, −4.49308755101533380061520889786, −4.16545178166705223662181081057, −3.79094395970967899686939313512, −3.33850054711156013910832543611, −3.00507224903418821482922074118, −1.81218389160224456671444937759, −1.65967377631536413091455379899, −0.845205837983328101174504416540,
0.845205837983328101174504416540, 1.65967377631536413091455379899, 1.81218389160224456671444937759, 3.00507224903418821482922074118, 3.33850054711156013910832543611, 3.79094395970967899686939313512, 4.16545178166705223662181081057, 4.49308755101533380061520889786, 4.89996582576554091537221458296, 5.78547542369211224423224850811, 5.79522770298325749593257454279, 5.82414629923919809239949169271, 6.46320083257403540216448331569, 6.71065097447877039768939551027, 7.08878816501079040308285094333, 7.80832452503619593074899096088, 8.017718260200687266378339625891, 8.081731312542757679368832380004, 8.600601626661742321236689991295, 8.846778275545068922937410317921, 8.974582059642785497132831744137, 9.421287975559228696494609430797, 9.937723593079180218837299839198, 10.36104760247380131767828863417, 10.55146886796298924031160393097
