| L(s) = 1 | − 40.2·3-s + 324.·5-s + 956.·7-s − 569.·9-s + 5.45e3·11-s + 6.28e3·13-s − 1.30e4·15-s + 3.45e4·17-s + 1.45e4·19-s − 3.84e4·21-s + 2.46e4·23-s + 2.71e4·25-s + 1.10e5·27-s − 1.71e5·29-s + 1.11e5·31-s − 2.19e5·33-s + 3.10e5·35-s − 1.03e5·37-s − 2.52e5·39-s − 7.16e4·41-s − 3.28e5·43-s − 1.84e5·45-s + 1.19e5·47-s + 9.22e4·49-s − 1.39e6·51-s − 1.04e6·53-s + 1.76e6·55-s + ⋯ |
| L(s) = 1 | − 0.859·3-s + 1.16·5-s + 1.05·7-s − 0.260·9-s + 1.23·11-s + 0.793·13-s − 0.998·15-s + 1.70·17-s + 0.488·19-s − 0.906·21-s + 0.422·23-s + 0.347·25-s + 1.08·27-s − 1.30·29-s + 0.673·31-s − 1.06·33-s + 1.22·35-s − 0.336·37-s − 0.682·39-s − 0.162·41-s − 0.629·43-s − 0.302·45-s + 0.167·47-s + 0.111·49-s − 1.46·51-s − 0.959·53-s + 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.797243474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.797243474\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 40.2T + 2.18e3T^{2} \) |
| 5 | \( 1 - 324.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 956.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.45e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.28e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.45e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.46e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.71e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.03e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.16e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.04e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.25e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.55e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.16e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.38e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.89e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.37e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.84e6T + 8.07e13T^{2} \) |
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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
−10.92904957214047270151471485297, −9.872875337639861574740849789737, −8.978567673770369480511514948816, −7.83723771730083851865955538538, −6.45574130489819968311438467740, −5.72279406800563351965766351199, −4.97406471342676743150286352638, −3.42953818577122992927252413283, −1.70416277914556560340084215358, −0.993452358297649226854781261236,
0.993452358297649226854781261236, 1.70416277914556560340084215358, 3.42953818577122992927252413283, 4.97406471342676743150286352638, 5.72279406800563351965766351199, 6.45574130489819968311438467740, 7.83723771730083851865955538538, 8.978567673770369480511514948816, 9.872875337639861574740849789737, 10.92904957214047270151471485297
