| L(s) = 1 | − 3.62·2-s − 27·3-s − 114.·4-s + 300.·5-s + 97.8·6-s + 1.35e3·7-s + 880.·8-s + 729·9-s − 1.08e3·10-s + 8.12e3·11-s + 3.10e3·12-s + 1.82e3·13-s − 4.90e3·14-s − 8.12e3·15-s + 1.15e4·16-s + 1.65e3·17-s − 2.64e3·18-s + 1.85e4·19-s − 3.45e4·20-s − 3.65e4·21-s − 2.94e4·22-s + 1.56e4·23-s − 2.37e4·24-s + 1.23e4·25-s − 6.60e3·26-s − 1.96e4·27-s − 1.55e5·28-s + ⋯ |
| L(s) = 1 | − 0.320·2-s − 0.577·3-s − 0.897·4-s + 1.07·5-s + 0.184·6-s + 1.49·7-s + 0.607·8-s + 0.333·9-s − 0.344·10-s + 1.84·11-s + 0.518·12-s + 0.230·13-s − 0.477·14-s − 0.621·15-s + 0.702·16-s + 0.0817·17-s − 0.106·18-s + 0.620·19-s − 0.965·20-s − 0.861·21-s − 0.589·22-s + 0.267·23-s − 0.350·24-s + 0.158·25-s − 0.0736·26-s − 0.192·27-s − 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.172650021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172650021\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 3.62T + 128T^{2} \) |
| 5 | \( 1 - 300.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.35e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 8.12e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.82e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.65e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.85e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.56e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.50e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.48e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.70e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.83e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.94e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.74e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.98e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 9.38e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.93e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.87e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.33e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.36e6T + 8.07e13T^{2} \) |
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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
−11.34096717351034137163768801291, −10.31314733770043313514916280783, −9.314147916830723058631057052752, −8.628405532918094697587541222153, −7.27750287755033187629063038951, −5.92433742548170768286310773914, −5.01778364131857629609010266795, −3.99112218723108361542920927329, −1.67144284774928147642343226479, −1.02026254511472121368843112605,
1.02026254511472121368843112605, 1.67144284774928147642343226479, 3.99112218723108361542920927329, 5.01778364131857629609010266795, 5.92433742548170768286310773914, 7.27750287755033187629063038951, 8.628405532918094697587541222153, 9.314147916830723058631057052752, 10.31314733770043313514916280783, 11.34096717351034137163768801291
