| L(s) = 1 | + 7.08·2-s + 18.2·4-s + 82.2·5-s − 97.4·8-s + 582.·10-s − 352.·11-s − 885.·13-s − 1.27e3·16-s + 425.·17-s − 1.56e3·19-s + 1.50e3·20-s − 2.49e3·22-s − 2.78e3·23-s + 3.63e3·25-s − 6.27e3·26-s − 3.67e3·29-s − 3.59e3·31-s − 5.92e3·32-s + 3.01e3·34-s + 1.42e4·37-s − 1.10e4·38-s − 8.00e3·40-s − 1.43e4·41-s + 7.58e3·43-s − 6.43e3·44-s − 1.97e4·46-s + 5.76e3·47-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.570·4-s + 1.47·5-s − 0.538·8-s + 1.84·10-s − 0.877·11-s − 1.45·13-s − 1.24·16-s + 0.356·17-s − 0.992·19-s + 0.839·20-s − 1.09·22-s − 1.09·23-s + 1.16·25-s − 1.82·26-s − 0.812·29-s − 0.671·31-s − 1.02·32-s + 0.447·34-s + 1.71·37-s − 1.24·38-s − 0.791·40-s − 1.33·41-s + 0.625·43-s − 0.500·44-s − 1.37·46-s + 0.380·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 7.08T + 32T^{2} \) |
| 5 | \( 1 - 82.2T + 3.12e3T^{2} \) |
| 11 | \( 1 + 352.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 885.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 425.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.58e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.76e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.53e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.37e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.75e4T + 8.58e9T^{2} \) |
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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
−9.881282553081270168388559443429, −9.233974166411197790143932034189, −7.87674852275254016306518420376, −6.65501976710982361956962249558, −5.72309790356314968275124045320, −5.21217200717232340297827561880, −4.15020303750955114707764145218, −2.68742590377036251162664939591, −2.06129429855006854656781456622, 0,
2.06129429855006854656781456622, 2.68742590377036251162664939591, 4.15020303750955114707764145218, 5.21217200717232340297827561880, 5.72309790356314968275124045320, 6.65501976710982361956962249558, 7.87674852275254016306518420376, 9.233974166411197790143932034189, 9.881282553081270168388559443429
