| L(s) = 1 | + 41.8·2-s − 81·3-s + 1.24e3·4-s + 372.·5-s − 3.39e3·6-s − 8.19e3·7-s + 3.05e4·8-s + 6.56e3·9-s + 1.55e4·10-s − 8.23e4·11-s − 1.00e5·12-s + 2.03e5·13-s − 3.43e5·14-s − 3.01e4·15-s + 6.43e5·16-s − 2.24e5·17-s + 2.74e5·18-s − 2.25e5·19-s + 4.62e5·20-s + 6.64e5·21-s − 3.44e6·22-s − 1.73e6·23-s − 2.47e6·24-s − 1.81e6·25-s + 8.52e6·26-s − 5.31e5·27-s − 1.01e7·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 0.577·3-s + 2.42·4-s + 0.266·5-s − 1.06·6-s − 1.29·7-s + 2.63·8-s + 0.333·9-s + 0.493·10-s − 1.69·11-s − 1.39·12-s + 1.97·13-s − 2.38·14-s − 0.153·15-s + 2.45·16-s − 0.651·17-s + 0.616·18-s − 0.396·19-s + 0.646·20-s + 0.745·21-s − 3.13·22-s − 1.28·23-s − 1.52·24-s − 0.928·25-s + 3.65·26-s − 0.192·27-s − 3.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 - 41.8T + 512T^{2} \) |
| 5 | \( 1 - 372.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.19e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.23e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.03e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.24e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.25e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.73e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.98e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.18e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.44e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.46e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.45e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.24e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.75e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 4.48e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.16e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.37e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.98e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.78e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.35e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.37e9T + 7.60e17T^{2} \) |
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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
−10.79971850144204604587957229300, −10.07431382531293151298438696808, −8.158193970659170403883534633596, −6.62898505000689170700321513408, −6.12996562181438395144941895898, −5.28830978047468504475037554670, −4.01757755332346036564560110655, −3.12563032345391349440706965887, −1.91558550631045613031247957128, 0,
1.91558550631045613031247957128, 3.12563032345391349440706965887, 4.01757755332346036564560110655, 5.28830978047468504475037554670, 6.12996562181438395144941895898, 6.62898505000689170700321513408, 8.158193970659170403883534633596, 10.07431382531293151298438696808, 10.79971850144204604587957229300
