| L(s) = 1 | − 3-s − 5-s + 2·7-s − 2·9-s + 7·11-s + 12·13-s + 15-s + 3·17-s + 4·19-s − 2·21-s + 8·23-s − 9·25-s + 3·27-s + 14·29-s + 9·31-s − 7·33-s − 2·35-s + 16·37-s − 12·39-s + 12·41-s + 43-s + 2·45-s − 47-s − 6·49-s − 3·51-s + 8·53-s − 7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s − 2/3·9-s + 2.11·11-s + 3.32·13-s + 0.258·15-s + 0.727·17-s + 0.917·19-s − 0.436·21-s + 1.66·23-s − 9/5·25-s + 0.577·27-s + 2.59·29-s + 1.61·31-s − 1.21·33-s − 0.338·35-s + 2.63·37-s − 1.92·39-s + 1.87·41-s + 0.152·43-s + 0.298·45-s − 0.145·47-s − 6/7·49-s − 0.420·51-s + 1.09·53-s − 0.943·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.14738880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.14738880\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{3} \) |
| 59 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + T + 2 p T^{2} + 12 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 10 T^{2} - 32 T^{3} + 10 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 34 T^{2} - 108 T^{3} + 34 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 12 T + 83 T^{2} - 361 T^{3} + 83 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 12 T^{2} + 44 T^{3} + 12 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 51 T^{2} - 195 T^{3} + 51 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 14 T + 147 T^{2} - 885 T^{3} + 147 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 16 T + 184 T^{2} - 34 p T^{3} + 184 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 128 T^{2} - 982 T^{3} + 128 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - T + 38 T^{2} - 354 T^{3} + 38 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + T + 122 T^{2} + 62 T^{3} + 122 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 174 T^{2} - 852 T^{3} + 174 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 152 T^{2} + 276 T^{3} + 152 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 13 T + 236 T^{2} - 1701 T^{3} + 236 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 33 T + 572 T^{2} - 5972 T^{3} + 572 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 34 T + 554 T^{2} - 5718 T^{3} + 554 p T^{4} - 34 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 17 T + 306 T^{2} + 2768 T^{3} + 306 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 3 T + 188 T^{2} + 499 T^{3} + 188 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 7 T + 198 T^{2} + 1134 T^{3} + 198 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 297 T^{2} - 1553 T^{3} + 297 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.89514276292334328264452458657, −6.43714713557526324060081089643, −6.43565031978008993098653358526, −6.36222304439479214489330863304, −5.90676147482548517398130223149, −5.82886069336218825590735420044, −5.81569162261955094424909777677, −5.07503794217253730958889470393, −5.04974390885259128736650549334, −4.99560872023952939355218639180, −4.37059911882205562547377921253, −4.27130793988416695565289466250, −4.03711978300717158974034311721, −3.71323948718281646445346331621, −3.66088943901783732223629265083, −3.39987813502841481963263614305, −2.96540851969244357965942388395, −2.69893785171636754524367371096, −2.50155000311126110492153967460, −2.02845403235604864580599491863, −1.56390881151499246092870649647, −1.08271803012120671727837519715, −1.02318353239968367828194634859, −0.887170769525716760418713689941, −0.73883906329654647818477679729,
0.73883906329654647818477679729, 0.887170769525716760418713689941, 1.02318353239968367828194634859, 1.08271803012120671727837519715, 1.56390881151499246092870649647, 2.02845403235604864580599491863, 2.50155000311126110492153967460, 2.69893785171636754524367371096, 2.96540851969244357965942388395, 3.39987813502841481963263614305, 3.66088943901783732223629265083, 3.71323948718281646445346331621, 4.03711978300717158974034311721, 4.27130793988416695565289466250, 4.37059911882205562547377921253, 4.99560872023952939355218639180, 5.04974390885259128736650549334, 5.07503794217253730958889470393, 5.81569162261955094424909777677, 5.82886069336218825590735420044, 5.90676147482548517398130223149, 6.36222304439479214489330863304, 6.43565031978008993098653358526, 6.43714713557526324060081089643, 6.89514276292334328264452458657
