| L(s) = 1 | + 8·2-s + 64.1·3-s + 64·4-s − 288.·5-s + 513.·6-s + 1.13e3·7-s + 512·8-s + 1.92e3·9-s − 2.30e3·10-s + 6.91e3·11-s + 4.10e3·12-s + 8.35e3·13-s + 9.09e3·14-s − 1.84e4·15-s + 4.09e3·16-s − 1.60e4·17-s + 1.54e4·18-s + 1.85e4·19-s − 1.84e4·20-s + 7.29e4·21-s + 5.53e4·22-s + 3.41e4·23-s + 3.28e4·24-s + 4.98e3·25-s + 6.68e4·26-s − 1.67e4·27-s + 7.27e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.37·3-s + 0.5·4-s − 1.03·5-s + 0.969·6-s + 1.25·7-s + 0.353·8-s + 0.880·9-s − 0.729·10-s + 1.56·11-s + 0.685·12-s + 1.05·13-s + 0.885·14-s − 1.41·15-s + 0.250·16-s − 0.791·17-s + 0.622·18-s + 0.619·19-s − 0.515·20-s + 1.71·21-s + 1.10·22-s + 0.585·23-s + 0.484·24-s + 0.0637·25-s + 0.745·26-s − 0.163·27-s + 0.626·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.458632802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.458632802\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 269 | \( 1 - 1.94e7T \) |
| good | 3 | \( 1 - 64.1T + 2.18e3T^{2} \) |
| 5 | \( 1 + 288.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.13e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.91e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.35e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.60e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.85e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.41e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.80e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.23e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.63e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.23e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.64e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.51e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.90e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.22e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.77e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 6.37e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.83e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.92e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.81e6T + 8.07e13T^{2} \) |
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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
−9.290604336243382862236249389259, −8.720134752356523361398541740424, −7.88577939933376008150199466666, −7.25485397904000869844688870902, −6.04652151754761743201002704101, −4.56869412541408041506587890174, −3.96009925249341220674984366973, −3.25016370829111510950547266462, −1.96933230367495089676940005686, −1.09254001688510598134945952326,
1.09254001688510598134945952326, 1.96933230367495089676940005686, 3.25016370829111510950547266462, 3.96009925249341220674984366973, 4.56869412541408041506587890174, 6.04652151754761743201002704101, 7.25485397904000869844688870902, 7.88577939933376008150199466666, 8.720134752356523361398541740424, 9.290604336243382862236249389259
