| L(s) = 1 | − 9·3-s − 69.8·5-s + 147.·7-s + 81·9-s − 294.·11-s − 1.11e3·13-s + 628.·15-s + 1.89e3·17-s − 361·19-s − 1.33e3·21-s − 828.·23-s + 1.75e3·25-s − 729·27-s + 1.43e3·29-s − 1.02e4·31-s + 2.65e3·33-s − 1.03e4·35-s + 1.42e4·37-s + 1.00e4·39-s − 1.37e4·41-s + 701.·43-s − 5.65e3·45-s + 3.56e3·47-s + 5.07e3·49-s − 1.70e4·51-s − 2.94e4·53-s + 2.05e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.24·5-s + 1.14·7-s + 0.333·9-s − 0.733·11-s − 1.82·13-s + 0.721·15-s + 1.59·17-s − 0.229·19-s − 0.658·21-s − 0.326·23-s + 0.560·25-s − 0.192·27-s + 0.317·29-s − 1.92·31-s + 0.423·33-s − 1.42·35-s + 1.71·37-s + 1.05·39-s − 1.27·41-s + 0.0578·43-s − 0.416·45-s + 0.235·47-s + 0.301·49-s − 0.918·51-s − 1.43·53-s + 0.916·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7185496794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7185496794\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 19 | \( 1 + 361T \) |
| good | 5 | \( 1 + 69.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 147.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 294.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.89e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 828.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.02e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.37e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 701.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.56e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.94e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.03e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.37e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.08e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.73e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.25e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.476686937765708534219278422262, −8.051539890774414030601048727174, −7.78192688407917461265393453668, −7.11030405443795111187365034674, −5.61715324944171958902494650470, −4.95543477296360769315175367153, −4.21936332544465385039572758238, −3.00866459229885942021109290571, −1.71355071877995380878807591149, −0.38526182465565005230588022823,
0.38526182465565005230588022823, 1.71355071877995380878807591149, 3.00866459229885942021109290571, 4.21936332544465385039572758238, 4.95543477296360769315175367153, 5.61715324944171958902494650470, 7.11030405443795111187365034674, 7.78192688407917461265393453668, 8.051539890774414030601048727174, 9.476686937765708534219278422262
