| L(s) = 1 | − 1.60·2-s + 208.·3-s − 509.·4-s + 625·5-s − 334.·6-s + 1.63e3·8-s + 2.38e4·9-s − 1.00e3·10-s − 1.92e4·11-s − 1.06e5·12-s − 5.24e4·13-s + 1.30e5·15-s + 2.58e5·16-s + 2.71e5·17-s − 3.81e4·18-s − 3.48e5·19-s − 3.18e5·20-s + 3.09e4·22-s − 8.21e5·23-s + 3.41e5·24-s + 3.90e5·25-s + 8.39e4·26-s + 8.63e5·27-s − 3.73e6·29-s − 2.08e5·30-s + 3.04e6·31-s − 1.25e6·32-s + ⋯ |
| L(s) = 1 | − 0.0708·2-s + 1.48·3-s − 0.994·4-s + 0.447·5-s − 0.105·6-s + 0.141·8-s + 1.21·9-s − 0.0316·10-s − 0.397·11-s − 1.47·12-s − 0.509·13-s + 0.664·15-s + 0.984·16-s + 0.787·17-s − 0.0857·18-s − 0.613·19-s − 0.444·20-s + 0.0281·22-s − 0.612·23-s + 0.210·24-s + 0.200·25-s + 0.0360·26-s + 0.312·27-s − 0.981·29-s − 0.0470·30-s + 0.592·31-s − 0.211·32-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 625T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.60T + 512T^{2} \) |
| 3 | \( 1 - 208.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 1.92e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.24e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.71e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.48e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 8.21e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.04e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.78e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.96e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 8.94e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.56e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.09e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.59e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.60e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.39e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.30e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 9.58e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.65e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.15e7T + 7.60e17T^{2} \) |
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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
−9.696720506186117484510127460466, −9.074757110851573046900567664121, −8.167240695764904768248302560982, −7.50912584777956230407343024357, −5.86765187557253580486270794404, −4.66777201561472470212210403207, −3.63096494454672504550448762911, −2.62926233305258452328408441436, −1.50795648064667472773408089066, 0,
1.50795648064667472773408089066, 2.62926233305258452328408441436, 3.63096494454672504550448762911, 4.66777201561472470212210403207, 5.86765187557253580486270794404, 7.50912584777956230407343024357, 8.167240695764904768248302560982, 9.074757110851573046900567664121, 9.696720506186117484510127460466
