L(s) = 1 | − 3-s − 16·5-s + 18·7-s − 31·9-s − 40·11-s + 95·13-s + 16·15-s − 42·17-s − 26·19-s − 18·21-s − 69·23-s + 157·25-s − 14·27-s + 181·29-s + 705·31-s + 40·33-s − 288·35-s + 80·37-s − 95·39-s + 281·41-s − 248·43-s + 496·45-s + 517·47-s − 369·49-s + 42·51-s + 190·53-s + 640·55-s + ⋯ |
L(s) = 1 | − 0.192·3-s − 1.43·5-s + 0.971·7-s − 1.14·9-s − 1.09·11-s + 2.02·13-s + 0.275·15-s − 0.599·17-s − 0.313·19-s − 0.187·21-s − 0.625·23-s + 1.25·25-s − 0.0997·27-s + 1.15·29-s + 4.08·31-s + 0.211·33-s − 1.39·35-s + 0.355·37-s − 0.390·39-s + 1.07·41-s − 0.879·43-s + 1.64·45-s + 1.60·47-s − 1.07·49-s + 0.115·51-s + 0.492·53-s + 1.56·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.104628610\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.104628610\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + p T )^{3} \) |
good | 3 | $S_4\times C_2$ | \( 1 + T + 32 T^{2} + 77 T^{3} + 32 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 16 T + 99 T^{2} - 232 T^{3} + 99 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 18 T + 99 p T^{2} - 10964 T^{3} + 99 p^{4} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 40 T + 1725 T^{2} + 40424 T^{3} + 1725 p^{3} T^{4} + 40 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 95 T + 7978 T^{2} - 385251 T^{3} + 7978 p^{3} T^{4} - 95 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 42 T + 12523 T^{2} + 356236 T^{3} + 12523 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 26 T + 1325 T^{2} - 787148 T^{3} + 1325 p^{3} T^{4} + 26 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 181 T + 32658 T^{2} - 1717337 T^{3} + 32658 p^{3} T^{4} - 181 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 705 T + 244660 T^{2} - 52242909 T^{3} + 244660 p^{3} T^{4} - 705 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 80 T + 28931 T^{2} + 11253064 T^{3} + 28931 p^{3} T^{4} - 80 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 281 T + 3494 p T^{2} - 41307101 T^{3} + 3494 p^{4} T^{4} - 281 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 248 T + 152377 T^{2} + 36661968 T^{3} + 152377 p^{3} T^{4} + 248 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 11 p T + 288700 T^{2} - 106705481 T^{3} + 288700 p^{3} T^{4} - 11 p^{7} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 190 T + 285155 T^{2} - 38423188 T^{3} + 285155 p^{3} T^{4} - 190 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 996 T + 829737 T^{2} + 392514584 T^{3} + 829737 p^{3} T^{4} + 996 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1214 T + 1069787 T^{2} - 564015572 T^{3} + 1069787 p^{3} T^{4} - 1214 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 776 T + 970373 T^{2} - 452560232 T^{3} + 970373 p^{3} T^{4} - 776 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 229 T + 917452 T^{2} - 164982785 T^{3} + 917452 p^{3} T^{4} - 229 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 333 T + 1019422 T^{2} + 267801129 T^{3} + 1019422 p^{3} T^{4} + 333 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 26 T + 1046041 T^{2} + 20149636 T^{3} + 1046041 p^{3} T^{4} + 26 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 474 T + 1539841 T^{2} - 550884324 T^{3} + 1539841 p^{3} T^{4} - 474 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1320 T + 1590475 T^{2} + 1206020752 T^{3} + 1590475 p^{3} T^{4} + 1320 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2386 T + 3755947 T^{2} - 3905686572 T^{3} + 3755947 p^{3} T^{4} - 2386 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.237425085007902047080101327433, −7.82199615019895581303515034606, −7.71955711647440220419497119394, −7.49958533632258940379596541152, −6.84745854138760102590960795811, −6.68529149947993891720702869995, −6.38515852651787941929973385490, −6.16466567356160506796398532773, −5.82203570554423947594225853644, −5.76438349573267361093507428935, −4.99913652983588003706459785330, −4.88289404396025635693092396452, −4.81091255969224530019384654138, −4.31959279849179437354974343203, −4.06957043727831851358021898624, −3.77778994704447044352732442110, −3.40605705360710064514985473825, −2.99684985243805051448208769333, −2.74641715741112312058985358223, −2.31350265054155103460987206531, −2.16880594163503479709839896277, −1.27723077123145478346334681322, −0.988955090345996178126488369965, −0.798678382865003062382215297000, −0.18693472396675260978476269932,
0.18693472396675260978476269932, 0.798678382865003062382215297000, 0.988955090345996178126488369965, 1.27723077123145478346334681322, 2.16880594163503479709839896277, 2.31350265054155103460987206531, 2.74641715741112312058985358223, 2.99684985243805051448208769333, 3.40605705360710064514985473825, 3.77778994704447044352732442110, 4.06957043727831851358021898624, 4.31959279849179437354974343203, 4.81091255969224530019384654138, 4.88289404396025635693092396452, 4.99913652983588003706459785330, 5.76438349573267361093507428935, 5.82203570554423947594225853644, 6.16466567356160506796398532773, 6.38515852651787941929973385490, 6.68529149947993891720702869995, 6.84745854138760102590960795811, 7.49958533632258940379596541152, 7.71955711647440220419497119394, 7.82199615019895581303515034606, 8.237425085007902047080101327433