| L(s) = 1 | + 11.1·2-s + 92.7·4-s + 10.1·5-s + 135.·7-s + 678.·8-s + 113.·10-s + 74.1·11-s − 252.·13-s + 1.51e3·14-s + 4.60e3·16-s − 233.·17-s + 550.·19-s + 941.·20-s + 827.·22-s − 1.95e3·23-s − 3.02e3·25-s − 2.81e3·26-s + 1.25e4·28-s + 4.67e3·29-s + 3.33e3·31-s + 2.97e4·32-s − 2.60e3·34-s + 1.37e3·35-s − 1.34e4·37-s + 6.14e3·38-s + 6.88e3·40-s − 8.20e3·41-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 2.89·4-s + 0.181·5-s + 1.04·7-s + 3.74·8-s + 0.358·10-s + 0.184·11-s − 0.414·13-s + 2.06·14-s + 4.49·16-s − 0.195·17-s + 0.349·19-s + 0.526·20-s + 0.364·22-s − 0.769·23-s − 0.967·25-s − 0.817·26-s + 3.03·28-s + 1.03·29-s + 0.622·31-s + 5.13·32-s − 0.386·34-s + 0.190·35-s − 1.62·37-s + 0.690·38-s + 0.680·40-s − 0.762·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.764416859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.764416859\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 - 11.1T + 32T^{2} \) |
| 5 | \( 1 - 10.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 135.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 74.1T + 1.61e5T^{2} \) |
| 13 | \( 1 + 252.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 233.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 550.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.95e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.20e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.34e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.49e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.91e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.26e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.23e4T + 8.58e9T^{2} \) |
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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
−10.87952085883945611367481153274, −10.05938818299495221904816785381, −8.270325016155472883000773240138, −7.35258583099012965269183692019, −6.39148026826924027030213376715, −5.40120355837174723721919701513, −4.67841110790830544841546442076, −3.72910286583776070102440598804, −2.45676124940303344029287221990, −1.51830862166337077648463722244,
1.51830862166337077648463722244, 2.45676124940303344029287221990, 3.72910286583776070102440598804, 4.67841110790830544841546442076, 5.40120355837174723721919701513, 6.39148026826924027030213376715, 7.35258583099012965269183692019, 8.270325016155472883000773240138, 10.05938818299495221904816785381, 10.87952085883945611367481153274
