| L(s) = 1 | − 8.18·2-s − 9·3-s + 35.0·4-s + 73.6·6-s + 163.·7-s − 24.5·8-s + 81·9-s − 121·11-s − 315.·12-s − 728.·13-s − 1.33e3·14-s − 918.·16-s − 1.47e3·17-s − 663.·18-s − 341.·19-s − 1.47e3·21-s + 990.·22-s − 2.94e3·23-s + 221.·24-s + 5.96e3·26-s − 729·27-s + 5.72e3·28-s − 2.82e3·29-s + 5.16e3·31-s + 8.30e3·32-s + 1.08e3·33-s + 1.20e4·34-s + ⋯ |
| L(s) = 1 | − 1.44·2-s − 0.577·3-s + 1.09·4-s + 0.835·6-s + 1.26·7-s − 0.135·8-s + 0.333·9-s − 0.301·11-s − 0.631·12-s − 1.19·13-s − 1.82·14-s − 0.897·16-s − 1.23·17-s − 0.482·18-s − 0.217·19-s − 0.728·21-s + 0.436·22-s − 1.16·23-s + 0.0784·24-s + 1.73·26-s − 0.192·27-s + 1.38·28-s − 0.623·29-s + 0.964·31-s + 1.43·32-s + 0.174·33-s + 1.79·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4290884812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4290884812\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 8.18T + 32T^{2} \) |
| 7 | \( 1 - 163.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 728.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.47e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 341.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.94e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.82e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.25e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.87e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.75e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.24e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.34e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.30e4T + 8.58e9T^{2} \) |
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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
−9.498973499944493568795340246805, −8.603189500050774268512270537751, −7.86166352090701694175797688675, −7.26467522991358564288711691417, −6.23697988148978293775697045596, −4.96898869440547700130906896149, −4.36734554318463253070638415890, −2.34740040754293296222443322203, −1.64258465624372901292740152231, −0.38285967482634164063268485726,
0.38285967482634164063268485726, 1.64258465624372901292740152231, 2.34740040754293296222443322203, 4.36734554318463253070638415890, 4.96898869440547700130906896149, 6.23697988148978293775697045596, 7.26467522991358564288711691417, 7.86166352090701694175797688675, 8.603189500050774268512270537751, 9.498973499944493568795340246805
