| L(s) = 1 | − 39.1·2-s + 81·3-s + 1.01e3·4-s + 1.84e3·5-s − 3.16e3·6-s + 8.71e3·7-s − 1.97e4·8-s + 6.56e3·9-s − 7.21e4·10-s + 1.15e4·11-s + 8.23e4·12-s + 8.11e3·13-s − 3.40e5·14-s + 1.49e5·15-s + 2.51e5·16-s − 2.58e5·17-s − 2.56e5·18-s − 5.11e5·19-s + 1.87e6·20-s + 7.06e5·21-s − 4.53e5·22-s + 1.50e6·23-s − 1.59e6·24-s + 1.45e6·25-s − 3.17e5·26-s + 5.31e5·27-s + 8.86e6·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 0.577·3-s + 1.98·4-s + 1.32·5-s − 0.997·6-s + 1.37·7-s − 1.70·8-s + 0.333·9-s − 2.28·10-s + 0.238·11-s + 1.14·12-s + 0.0787·13-s − 2.37·14-s + 0.762·15-s + 0.959·16-s − 0.751·17-s − 0.576·18-s − 0.900·19-s + 2.62·20-s + 0.792·21-s − 0.412·22-s + 1.12·23-s − 0.984·24-s + 0.743·25-s − 0.136·26-s + 0.192·27-s + 2.72·28-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
| \(L(5)\) |
\(\approx\) |
\(2.042903166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.042903166\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 39.1T + 512T^{2} \) |
| 5 | \( 1 - 1.84e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.71e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.15e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 8.11e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.58e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.11e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.50e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.60e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.17e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.37e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.04e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.15e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.05e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.04e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 1.74e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.95e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.53e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.49e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.98e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.09e6T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.94e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.22e8T + 7.60e17T^{2} \) |
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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
−10.69410201661977182530100731102, −9.815862787996613911786845341148, −8.894661277770774302736041756758, −8.386282078757159749321237859854, −7.24550518327418508703405923684, −6.20959173784272378650857970265, −4.68459223846610788841606420706, −2.51041637421155693376144076723, −1.81535654463232563533935239719, −0.946798812473760523455292331044,
0.946798812473760523455292331044, 1.81535654463232563533935239719, 2.51041637421155693376144076723, 4.68459223846610788841606420706, 6.20959173784272378650857970265, 7.24550518327418508703405923684, 8.386282078757159749321237859854, 8.894661277770774302736041756758, 9.815862787996613911786845341148, 10.69410201661977182530100731102
