| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 196.·5-s + 216·6-s − 343·7-s + 512·8-s + 729·9-s + 1.56e3·10-s − 2.84e3·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s + 5.29e3·15-s + 4.09e3·16-s + 1.61e3·17-s + 5.83e3·18-s + 5.58e4·19-s + 1.25e4·20-s − 9.26e3·21-s − 2.27e4·22-s + 2.85e4·23-s + 1.38e4·24-s − 3.96e4·25-s − 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.702·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.496·10-s − 0.643·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.405·15-s + 0.250·16-s + 0.0796·17-s + 0.235·18-s + 1.86·19-s + 0.351·20-s − 0.218·21-s − 0.455·22-s + 0.488·23-s + 0.204·24-s − 0.507·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.491112433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.491112433\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 5 | \( 1 - 196.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 2.84e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.61e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.58e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.85e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.12e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.22e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.63e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.24e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.85e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.02e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.71e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.81e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.08e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.76e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 9.89e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.50e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.05e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.83e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.31e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.91e6T + 8.07e13T^{2} \) |
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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
−9.749534205097660295566229207246, −8.893461518636075004234112597092, −7.64842061505265208063045559199, −7.03582528340023881367505524309, −5.76542568053199120960750614615, −5.19754782557209398767154064059, −3.89346967910302961449341238277, −2.93965337575013237716109965655, −2.15394715295322917362626990180, −0.892469085629176121408280547779,
0.892469085629176121408280547779, 2.15394715295322917362626990180, 2.93965337575013237716109965655, 3.89346967910302961449341238277, 5.19754782557209398767154064059, 5.76542568053199120960750614615, 7.03582528340023881367505524309, 7.64842061505265208063045559199, 8.893461518636075004234112597092, 9.749534205097660295566229207246
