| L(s) = 1 | + 8·2-s + 47.3·3-s + 64·4-s + 499.·5-s + 378.·6-s + 1.39e3·7-s + 512·8-s + 55.9·9-s + 3.99e3·10-s + 3.03e3·12-s + 471.·13-s + 1.11e4·14-s + 2.36e4·15-s + 4.09e3·16-s − 1.77e4·17-s + 447.·18-s + 3.74e4·19-s + 3.19e4·20-s + 6.59e4·21-s − 3.14e4·23-s + 2.42e4·24-s + 1.71e5·25-s + 3.76e3·26-s − 1.00e5·27-s + 8.90e4·28-s + 1.61e5·29-s + 1.89e5·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.01·3-s + 0.5·4-s + 1.78·5-s + 0.716·6-s + 1.53·7-s + 0.353·8-s + 0.0255·9-s + 1.26·10-s + 0.506·12-s + 0.0594·13-s + 1.08·14-s + 1.80·15-s + 0.250·16-s − 0.874·17-s + 0.0180·18-s + 1.25·19-s + 0.893·20-s + 1.55·21-s − 0.539·23-s + 0.358·24-s + 2.19·25-s + 0.0420·26-s − 0.986·27-s + 0.766·28-s + 1.22·29-s + 1.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.073161558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.073161558\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 47.3T + 2.18e3T^{2} \) |
| 5 | \( 1 - 499.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.39e3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 471.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.77e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.74e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.14e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.61e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.65e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.76e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.46e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.25e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.78e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.03e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.88e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.46e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.85e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.90e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.92e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.61e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.74e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.02e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.21e7T + 8.07e13T^{2} \) |
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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
−10.88422222053938004173938637929, −9.841224145401184788693555094583, −8.885944419443715638982840806964, −8.028368496224458418611463218979, −6.74168171457874536943109983798, −5.52212073583887239779009312785, −4.84668551283013680333322875385, −3.24145793411433000006887350823, −2.08146526989796960440133903584, −1.58821490397184076816137052163,
1.58821490397184076816137052163, 2.08146526989796960440133903584, 3.24145793411433000006887350823, 4.84668551283013680333322875385, 5.52212073583887239779009312785, 6.74168171457874536943109983798, 8.028368496224458418611463218979, 8.885944419443715638982840806964, 9.841224145401184788693555094583, 10.88422222053938004173938637929
