| L(s) = 1 | − 195.·3-s − 426.·5-s + 2.40e3·7-s + 1.84e4·9-s + 4.48e4·11-s + 3.04e4·13-s + 8.32e4·15-s + 1.47e5·17-s + 2.35e5·19-s − 4.69e5·21-s − 3.61e5·23-s − 1.77e6·25-s + 2.33e5·27-s − 7.37e6·29-s + 6.55e5·31-s − 8.76e6·33-s − 1.02e6·35-s − 9.92e6·37-s − 5.95e6·39-s + 7.31e6·41-s − 2.82e6·43-s − 7.87e6·45-s + 1.25e6·47-s + 5.76e6·49-s − 2.88e7·51-s − 3.49e7·53-s − 1.91e7·55-s + ⋯ |
| L(s) = 1 | − 1.39·3-s − 0.304·5-s + 0.377·7-s + 0.939·9-s + 0.923·11-s + 0.295·13-s + 0.424·15-s + 0.428·17-s + 0.414·19-s − 0.526·21-s − 0.269·23-s − 0.907·25-s + 0.0845·27-s − 1.93·29-s + 0.127·31-s − 1.28·33-s − 0.115·35-s − 0.870·37-s − 0.411·39-s + 0.404·41-s − 0.126·43-s − 0.286·45-s + 0.0376·47-s + 0.142·49-s − 0.597·51-s − 0.607·53-s − 0.281·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 + 195.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 426.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 4.48e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.04e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.47e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.35e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.61e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.37e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.55e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.92e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.31e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.82e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.25e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.49e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.71e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.08e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.28e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.96e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.26e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.36e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.60e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.09e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.20e8T + 7.60e17T^{2} \) |
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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
−12.34416774880327130849055257873, −11.60786266392310962449876735203, −10.78054273681240517429252717712, −9.354082384933849216768465155701, −7.69103131378048265940759260152, −6.33826789216900189043523179850, −5.28398594976750374664568418502, −3.86108599252687704605962003709, −1.42662743739291357827395631783, 0,
1.42662743739291357827395631783, 3.86108599252687704605962003709, 5.28398594976750374664568418502, 6.33826789216900189043523179850, 7.69103131378048265940759260152, 9.354082384933849216768465155701, 10.78054273681240517429252717712, 11.60786266392310962449876735203, 12.34416774880327130849055257873
