| L(s) = 1 | + 29.5·2-s − 81·3-s + 363.·4-s + 625·5-s − 2.39e3·6-s + 5.26e3·7-s − 4.38e3·8-s + 6.56e3·9-s + 1.84e4·10-s − 1.66e4·11-s − 2.94e4·12-s + 8.79e4·13-s + 1.55e5·14-s − 5.06e4·15-s − 3.16e5·16-s − 3.89e4·17-s + 1.94e5·18-s − 1.30e5·19-s + 2.27e5·20-s − 4.26e5·21-s − 4.93e5·22-s + 1.43e6·23-s + 3.55e5·24-s + 3.90e5·25-s + 2.60e6·26-s − 5.31e5·27-s + 1.91e6·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s − 0.577·3-s + 0.710·4-s + 0.447·5-s − 0.755·6-s + 0.828·7-s − 0.378·8-s + 0.333·9-s + 0.584·10-s − 0.343·11-s − 0.410·12-s + 0.854·13-s + 1.08·14-s − 0.258·15-s − 1.20·16-s − 0.113·17-s + 0.435·18-s − 0.229·19-s + 0.317·20-s − 0.478·21-s − 0.449·22-s + 1.07·23-s + 0.218·24-s + 0.200·25-s + 1.11·26-s − 0.192·27-s + 0.588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.477683446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.477683446\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 - 29.5T + 512T^{2} \) |
| 7 | \( 1 - 5.26e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.66e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 8.79e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.89e4T + 1.18e11T^{2} \) |
| 23 | \( 1 - 1.43e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.99e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.55e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.20e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.52e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 5.25e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.40e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.99e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.20e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.56e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.29e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.84e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.55e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.70e8T + 7.60e17T^{2} \) |
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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
−10.73156811594032800367877841341, −9.329872998023637384822318542238, −8.295775243263798023233320502668, −6.90970752043932867864393460553, −6.01833488180238929283498272691, −5.17940715088528523847056222143, −4.50054029599407136084957370554, −3.32126558136360862884377832767, −2.07757087460644398341736806378, −0.811120102994213745025647786775,
0.811120102994213745025647786775, 2.07757087460644398341736806378, 3.32126558136360862884377832767, 4.50054029599407136084957370554, 5.17940715088528523847056222143, 6.01833488180238929283498272691, 6.90970752043932867864393460553, 8.295775243263798023233320502668, 9.329872998023637384822318542238, 10.73156811594032800367877841341
