| L(s) = 1 | + 27.0·2-s + 81·3-s + 222.·4-s − 1.51e3·5-s + 2.19e3·6-s + 2.21e3·7-s − 7.84e3·8-s + 6.56e3·9-s − 4.11e4·10-s + 2.04e4·11-s + 1.80e4·12-s + 1.46e5·13-s + 5.98e4·14-s − 1.22e5·15-s − 3.26e5·16-s + 1.06e4·17-s + 1.77e5·18-s − 3.63e5·19-s − 3.37e5·20-s + 1.79e5·21-s + 5.55e5·22-s + 1.47e6·23-s − 6.35e5·24-s + 3.51e5·25-s + 3.97e6·26-s + 5.31e5·27-s + 4.91e5·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 0.577·3-s + 0.434·4-s − 1.08·5-s + 0.691·6-s + 0.347·7-s − 0.677·8-s + 0.333·9-s − 1.30·10-s + 0.421·11-s + 0.250·12-s + 1.42·13-s + 0.416·14-s − 0.627·15-s − 1.24·16-s + 0.0309·17-s + 0.399·18-s − 0.639·19-s − 0.471·20-s + 0.200·21-s + 0.505·22-s + 1.10·23-s − 0.391·24-s + 0.179·25-s + 1.70·26-s + 0.192·27-s + 0.151·28-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 | \( 1 - 27.0T + 512T^{2} \) |
| 5 | \( 1 + 1.51e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.21e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.04e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.46e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.06e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.63e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.47e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.12e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.92e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.34e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.67e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.05e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.26e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 3.72e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.12e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.42e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.68e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.59e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.22e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.52e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.40e8T + 7.60e17T^{2} \) |
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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
−11.05978530931261986640002503569, −9.235307592222706752536316548036, −8.475298770335915483788475701648, −7.35571859904154577793336377002, −6.15061961494707400355735412344, −4.88088408837944115247741948899, −3.79367108168903866135994865499, −3.39504897339055304746064896815, −1.68357125683118548993944331613, 0,
1.68357125683118548993944331613, 3.39504897339055304746064896815, 3.79367108168903866135994865499, 4.88088408837944115247741948899, 6.15061961494707400355735412344, 7.35571859904154577793336377002, 8.475298770335915483788475701648, 9.235307592222706752536316548036, 11.05978530931261986640002503569
