| L(s) = 1 | + 3.38·2-s − 9·3-s − 20.5·4-s + 54.5·5-s − 30.4·6-s − 177.·8-s + 81·9-s + 184.·10-s + 481.·11-s + 185.·12-s − 512.·13-s − 490.·15-s + 57.6·16-s − 590.·17-s + 273.·18-s − 2.45e3·19-s − 1.12e3·20-s + 1.62e3·22-s + 1.77e3·23-s + 1.59e3·24-s − 151.·25-s − 1.73e3·26-s − 729·27-s − 4.24e3·29-s − 1.65e3·30-s − 9.76e3·31-s + 5.88e3·32-s + ⋯ |
| L(s) = 1 | + 0.597·2-s − 0.577·3-s − 0.642·4-s + 0.975·5-s − 0.345·6-s − 0.981·8-s + 0.333·9-s + 0.582·10-s + 1.20·11-s + 0.371·12-s − 0.841·13-s − 0.563·15-s + 0.0562·16-s − 0.495·17-s + 0.199·18-s − 1.55·19-s − 0.627·20-s + 0.717·22-s + 0.699·23-s + 0.566·24-s − 0.0486·25-s − 0.502·26-s − 0.192·27-s − 0.937·29-s − 0.336·30-s − 1.82·31-s + 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3.38T + 32T^{2} \) |
| 5 | \( 1 - 54.5T + 3.12e3T^{2} \) |
| 11 | \( 1 - 481.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 512.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 590.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.45e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.77e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.32e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.48e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.65e4T + 8.58e9T^{2} \) |
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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
−11.87778343185527530878441339498, −10.64470567780617521875461009684, −9.505084519057962723090913644633, −8.834158993958807835694063422299, −6.92697378147176853223863856295, −5.94598370385881742327189243980, −4.97469035891543668449611023766, −3.82123450610908575845715636795, −1.89718704709766317081605704997, 0,
1.89718704709766317081605704997, 3.82123450610908575845715636795, 4.97469035891543668449611023766, 5.94598370385881742327189243980, 6.92697378147176853223863856295, 8.834158993958807835694063422299, 9.505084519057962723090913644633, 10.64470567780617521875461009684, 11.87778343185527530878441339498
