| L(s) = 1 | − 4·2-s + 16·4-s − 78.4·5-s + 221.·7-s − 64·8-s + 313.·10-s − 230.·11-s − 771.·13-s − 884.·14-s + 256·16-s + 769.·17-s − 383.·19-s − 1.25e3·20-s + 921.·22-s − 386.·23-s + 3.03e3·25-s + 3.08e3·26-s + 3.53e3·28-s + 789.·29-s + 3.21e3·31-s − 1.02e3·32-s − 3.07e3·34-s − 1.73e4·35-s + 2.46e3·37-s + 1.53e3·38-s + 5.02e3·40-s + 9.24e3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.40·5-s + 1.70·7-s − 0.353·8-s + 0.992·10-s − 0.573·11-s − 1.26·13-s − 1.20·14-s + 0.250·16-s + 0.646·17-s − 0.243·19-s − 0.701·20-s + 0.405·22-s − 0.152·23-s + 0.971·25-s + 0.895·26-s + 0.852·28-s + 0.174·29-s + 0.601·31-s − 0.176·32-s − 0.456·34-s − 2.39·35-s + 0.296·37-s + 0.172·38-s + 0.496·40-s + 0.858·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.062814199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.062814199\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 78.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 221.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 230.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 771.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 769.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 383.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 386.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 789.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.24e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.95e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.25e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.38e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.28e4T + 8.58e9T^{2} \) |
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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
−11.73017295564085034593139381267, −11.08221297424151684742979268439, −10.03421423309559206175615791339, −8.528272394411684349713820943890, −7.85787383432893677848582327143, −7.27016633868331263317537551162, −5.27370226182943070499330047453, −4.19428576380716705474695119111, −2.41211379689214149845987506723, −0.73847098266102616204954202130,
0.73847098266102616204954202130, 2.41211379689214149845987506723, 4.19428576380716705474695119111, 5.27370226182943070499330047453, 7.27016633868331263317537551162, 7.85787383432893677848582327143, 8.528272394411684349713820943890, 10.03421423309559206175615791339, 11.08221297424151684742979268439, 11.73017295564085034593139381267
