| L(s) = 1 | + 5.15·2-s + 3·3-s + 18.5·4-s + 1.69·5-s + 15.4·6-s − 11.0·7-s + 54.3·8-s + 9·9-s + 8.71·10-s − 10.1·11-s + 55.6·12-s + 29.5·13-s − 57.1·14-s + 5.07·15-s + 131.·16-s − 29.3·17-s + 46.3·18-s − 46.9·19-s + 31.3·20-s − 33.2·21-s − 52.2·22-s + 22.2·23-s + 163.·24-s − 122.·25-s + 152.·26-s + 27·27-s − 205.·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 0.577·3-s + 2.31·4-s + 0.151·5-s + 1.05·6-s − 0.599·7-s + 2.40·8-s + 0.333·9-s + 0.275·10-s − 0.277·11-s + 1.33·12-s + 0.630·13-s − 1.09·14-s + 0.0873·15-s + 2.05·16-s − 0.419·17-s + 0.607·18-s − 0.566·19-s + 0.350·20-s − 0.345·21-s − 0.506·22-s + 0.201·23-s + 1.38·24-s − 0.977·25-s + 1.14·26-s + 0.192·27-s − 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.579728780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.579728780\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 5.15T + 8T^{2} \) |
| 5 | \( 1 - 1.69T + 125T^{2} \) |
| 7 | \( 1 + 11.0T + 343T^{2} \) |
| 11 | \( 1 + 10.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 52.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.51T + 5.06e4T^{2} \) |
| 41 | \( 1 - 201.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 479.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 133.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 484.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 578.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 52.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 399.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 269.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 174.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 628.T + 9.12e5T^{2} \) |
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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
−12.79459258131240119002766808175, −11.52216483741965903971549933484, −10.61069309396721413978549342556, −9.265800223108257562482672059304, −7.76516664233551999785418742273, −6.59413564132328852160814775379, −5.71232996827151363675249787305, −4.33758419198079730335547234243, −3.36664135883380148032726680006, −2.14835795249272276993995709581,
2.14835795249272276993995709581, 3.36664135883380148032726680006, 4.33758419198079730335547234243, 5.71232996827151363675249787305, 6.59413564132328852160814775379, 7.76516664233551999785418742273, 9.265800223108257562482672059304, 10.61069309396721413978549342556, 11.52216483741965903971549933484, 12.79459258131240119002766808175
