| L(s) = 1 | − 38.1·2-s + 944.·4-s + 109.·5-s + 5.94e3·7-s − 1.65e4·8-s − 4.18e3·10-s + 2.52e4·11-s + 2.85e4·13-s − 2.27e5·14-s + 1.46e5·16-s − 1.09e5·17-s − 9.04e5·19-s + 1.03e5·20-s − 9.62e5·22-s + 4.35e5·23-s − 1.94e6·25-s − 1.09e6·26-s + 5.61e6·28-s − 6.44e6·29-s + 6.62e6·31-s + 2.85e6·32-s + 4.17e6·34-s + 6.52e5·35-s + 4.14e6·37-s + 3.45e7·38-s − 1.81e6·40-s − 1.49e7·41-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.84·4-s + 0.0785·5-s + 0.936·7-s − 1.42·8-s − 0.132·10-s + 0.519·11-s + 0.277·13-s − 1.57·14-s + 0.560·16-s − 0.317·17-s − 1.59·19-s + 0.144·20-s − 0.875·22-s + 0.324·23-s − 0.993·25-s − 0.467·26-s + 1.72·28-s − 1.69·29-s + 1.28·31-s + 0.481·32-s + 0.535·34-s + 0.0735·35-s + 0.363·37-s + 2.68·38-s − 0.111·40-s − 0.826·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 + 38.1T + 512T^{2} \) |
| 5 | \( 1 - 109.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.94e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.52e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 1.09e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 4.35e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.62e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.14e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.49e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.01e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.30e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.53e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.52e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.66e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.01e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.13e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.34e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.47e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.25e9T + 7.60e17T^{2} \) |
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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
−11.04734909755628298571923022229, −10.06461470903940030393010402941, −8.963952056369757456428087777344, −8.262423814090666430863202812743, −7.24474218217664709074168206206, −6.05944702904612976942238715271, −4.27100463854319484887376373174, −2.25134408688810868612373012497, −1.33503831324259175371003039560, 0,
1.33503831324259175371003039560, 2.25134408688810868612373012497, 4.27100463854319484887376373174, 6.05944702904612976942238715271, 7.24474218217664709074168206206, 8.262423814090666430863202812743, 8.963952056369757456428087777344, 10.06461470903940030393010402941, 11.04734909755628298571923022229
