| L(s) = 1 | − 34.1·2-s − 128.·3-s + 652.·4-s + 625·5-s + 4.38e3·6-s − 4.80e3·8-s − 3.21e3·9-s − 2.13e4·10-s − 2.66e4·11-s − 8.37e4·12-s + 1.32e5·13-s − 8.02e4·15-s − 1.70e5·16-s − 6.18e5·17-s + 1.09e5·18-s + 8.29e5·19-s + 4.07e5·20-s + 9.11e5·22-s − 1.47e6·23-s + 6.16e5·24-s + 3.90e5·25-s − 4.53e6·26-s + 2.93e6·27-s − 2.50e6·29-s + 2.73e6·30-s + 7.91e6·31-s + 8.27e6·32-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 0.914·3-s + 1.27·4-s + 0.447·5-s + 1.37·6-s − 0.414·8-s − 0.163·9-s − 0.674·10-s − 0.549·11-s − 1.16·12-s + 1.29·13-s − 0.409·15-s − 0.649·16-s − 1.79·17-s + 0.246·18-s + 1.46·19-s + 0.570·20-s + 0.829·22-s − 1.10·23-s + 0.379·24-s + 0.200·25-s − 1.94·26-s + 1.06·27-s − 0.656·29-s + 0.617·30-s + 1.53·31-s + 1.39·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5080814337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5080814337\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 625T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 34.1T + 512T^{2} \) |
| 3 | \( 1 + 128.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 2.66e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.32e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.18e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.29e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.47e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.50e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.91e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.58e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.61e5T + 3.27e14T^{2} \) |
| 43 | \( 1 - 5.00e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.34e5T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.05e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.22e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.02e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.08e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 7.47e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.10e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.30e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.85e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.72e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.03e8T + 7.60e17T^{2} \) |
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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
−10.47819901182460825531291975075, −9.514562608010355661716224573751, −8.663133059683176481667889887434, −7.77213109210290031553506264357, −6.53867701780348637827830992024, −5.84887344891189691512550031026, −4.51298218211041816606222639399, −2.66805699882523536228795738023, −1.42777122827974379080456778108, −0.46265917565060696656456377019,
0.46265917565060696656456377019, 1.42777122827974379080456778108, 2.66805699882523536228795738023, 4.51298218211041816606222639399, 5.84887344891189691512550031026, 6.53867701780348637827830992024, 7.77213109210290031553506264357, 8.663133059683176481667889887434, 9.514562608010355661716224573751, 10.47819901182460825531291975075
