L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 356.·5-s − 216·6-s − 343·7-s + 512·8-s + 729·9-s − 2.84e3·10-s − 7.50e3·11-s − 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s + 9.61e3·15-s + 4.09e3·16-s − 1.60e4·17-s + 5.83e3·18-s + 2.77e3·19-s − 2.27e4·20-s + 9.26e3·21-s − 6.00e4·22-s − 3.25e4·23-s − 1.38e4·24-s + 4.87e4·25-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 − 1.27·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.901·10-s − 1.70·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.735·15-s + 0.250·16-s − 0.793·17-s + 0.235·18-s + 0.0928·19-s − 0.637·20-s + 0.218·21-s − 1.20·22-s − 0.557·23-s − 0.204·24-s + 0.623·25-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
\(L(4)\) |
\(\approx\) |
\(0.3462922357\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3462922357\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 + 356.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 7.50e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.60e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.77e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.25e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.68e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.20e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.11e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.24e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.58e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.72e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.81e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.08e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.27e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.65e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.18e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.50e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.44e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.97e6T + 8.07e13T^{2} \) |
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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
−9.942664464174491918668738315507, −8.556435874851629467516550332216, −7.63675121230616461357800121170, −7.01898570636203708797180360567, −5.81846604550731976363888166790, −5.04300767630174505864343780699, −4.04855794116770796142311025942, −3.23940837863509245379639811319, −1.97185202713545342121048899135, −0.22019230176235425505212316589,
0.22019230176235425505212316589, 1.97185202713545342121048899135, 3.23940837863509245379639811319, 4.04855794116770796142311025942, 5.04300767630174505864343780699, 5.81846604550731976363888166790, 7.01898570636203708797180360567, 7.63675121230616461357800121170, 8.556435874851629467516550332216, 9.942664464174491918668738315507