| L(s) = 1 | − 16·2-s + 115.·3-s + 256·4-s − 1.19e3·5-s − 1.85e3·6-s + 1.10e4·7-s − 4.09e3·8-s − 6.30e3·9-s + 1.90e4·10-s − 1.66e4·11-s + 2.96e4·12-s − 1.76e5·14-s − 1.38e5·15-s + 6.55e4·16-s + 1.58e4·17-s + 1.00e5·18-s − 8.33e4·19-s − 3.05e5·20-s + 1.27e6·21-s + 2.66e5·22-s + 2.11e6·23-s − 4.73e5·24-s − 5.28e5·25-s − 3.00e6·27-s + 2.81e6·28-s + 9.16e5·29-s + 2.20e6·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.824·3-s + 0.5·4-s − 0.854·5-s − 0.582·6-s + 1.73·7-s − 0.353·8-s − 0.320·9-s + 0.603·10-s − 0.343·11-s + 0.412·12-s − 1.22·14-s − 0.704·15-s + 0.250·16-s + 0.0460·17-s + 0.226·18-s − 0.146·19-s − 0.427·20-s + 1.42·21-s + 0.242·22-s + 1.57·23-s − 0.291·24-s − 0.270·25-s − 1.08·27-s + 0.866·28-s + 0.240·29-s + 0.497·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 16T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 115.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.19e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.10e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.66e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 1.58e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.33e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.11e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.16e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.29e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.39e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.82e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.44e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.99e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.33e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.58e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.91e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.87e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.72e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.30e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.94e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.08e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.55e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.83e8T + 7.60e17T^{2} \) |
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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
−9.099859814121440649312507422574, −8.609325250306582305804048633968, −7.74287286787823026422178204310, −7.35163241880435858355822479344, −5.60511528677607822334240743606, −4.53320501432840356388869818159, −3.34259268532672458510684347424, −2.25864998266568267626209216397, −1.27874970938803711402798858810, 0,
1.27874970938803711402798858810, 2.25864998266568267626209216397, 3.34259268532672458510684347424, 4.53320501432840356388869818159, 5.60511528677607822334240743606, 7.35163241880435858355822479344, 7.74287286787823026422178204310, 8.609325250306582305804048633968, 9.099859814121440649312507422574
