| L(s) = 1 | + 2·3-s + 96·5-s + 243·9-s + 720·11-s + 1.14e3·13-s + 192·15-s − 1.25e3·17-s + 94·19-s − 96·23-s + 3.12e3·25-s + 1.45e3·27-s − 8.74e3·29-s + 6.24e3·31-s + 1.44e3·33-s + 1.07e4·37-s + 2.28e3·39-s + 2.40e4·41-s − 1.83e4·43-s + 2.33e4·45-s + 2.58e4·47-s − 2.50e3·51-s − 1.01e3·53-s + 6.91e4·55-s + 188·57-s − 1.24e3·59-s − 7.59e3·61-s + 1.09e5·65-s + ⋯ |
| L(s) = 1 | + 0.128·3-s + 1.71·5-s + 9-s + 1.79·11-s + 1.87·13-s + 0.220·15-s − 1.05·17-s + 0.0597·19-s − 0.0378·23-s + 25-s + 0.382·27-s − 1.93·29-s + 1.16·31-s + 0.230·33-s + 1.29·37-s + 0.240·39-s + 2.23·41-s − 1.51·43-s + 1.71·45-s + 1.70·47-s − 0.135·51-s − 0.0495·53-s + 3.08·55-s + 0.00766·57-s − 0.0464·59-s − 0.261·61-s + 3.22·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.981212346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.981212346\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T - 239 T^{2} - 2 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 96 T + 6091 T^{2} - 96 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 720 T + 357349 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 44 p T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1254 T + 152659 T^{2} + 1254 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 94 T - 2467263 T^{2} - 94 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 96 T - 6427127 T^{2} + 96 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4374 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 6244 T + 10358385 T^{2} - 6244 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10798 T + 47252847 T^{2} - 10798 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12006 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9160 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 25836 T + 438153889 T^{2} - 25836 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1014 T - 417167297 T^{2} + 1014 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 1242 T - 713381735 T^{2} + 1242 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7592 T - 786957837 T^{2} + 7592 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 41132 T + 341716317 T^{2} + 41132 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 37632 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 13438 T - 1892491749 T^{2} - 13438 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6248 T - 3038018895 T^{2} + 6248 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 25254 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 45126 T - 3547703573 T^{2} - 45126 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 107222 T + p^{5} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.69892028731118300029891720455, −11.32464509502692729273788668847, −10.94867083935413610935020352043, −10.11822321942125499940221524956, −10.00834919721597987986457834928, −9.246517589269486315574353788281, −8.903823049369117879779130538055, −8.816814517174552640553453559451, −7.66335222581771956890716831223, −7.23787286664908900574329845401, −6.34816407409272334377585672761, −6.14327969513421458966312187946, −5.96517132056709824531322519630, −4.88413329179933057710171367319, −4.00948766188503505360097222881, −3.96232046359291411778283530961, −2.76775958912000936925935696216, −1.89389763388040805378306134700, −1.45585746379722354554727224778, −0.895393808723037845062778731143,
0.895393808723037845062778731143, 1.45585746379722354554727224778, 1.89389763388040805378306134700, 2.76775958912000936925935696216, 3.96232046359291411778283530961, 4.00948766188503505360097222881, 4.88413329179933057710171367319, 5.96517132056709824531322519630, 6.14327969513421458966312187946, 6.34816407409272334377585672761, 7.23787286664908900574329845401, 7.66335222581771956890716831223, 8.816814517174552640553453559451, 8.903823049369117879779130538055, 9.246517589269486315574353788281, 10.00834919721597987986457834928, 10.11822321942125499940221524956, 10.94867083935413610935020352043, 11.32464509502692729273788668847, 11.69892028731118300029891720455
