| L(s) = 1 | − 8·2-s + 91·3-s + 64·4-s + 185·5-s − 728·6-s − 722·7-s − 512·8-s + 6.09e3·9-s − 1.48e3·10-s − 1.33e3·11-s + 5.82e3·12-s + 1.10e4·13-s + 5.77e3·14-s + 1.68e4·15-s + 4.09e3·16-s − 1.72e4·17-s − 4.87e4·18-s − 9.28e3·19-s + 1.18e4·20-s − 6.57e4·21-s + 1.06e4·22-s + 2.29e4·23-s − 4.65e4·24-s − 4.39e4·25-s − 8.81e4·26-s + 3.55e5·27-s − 4.62e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.94·3-s + 1/2·4-s + 0.661·5-s − 1.37·6-s − 0.795·7-s − 0.353·8-s + 2.78·9-s − 0.468·10-s − 0.301·11-s + 0.972·12-s + 1.39·13-s + 0.562·14-s + 1.28·15-s + 1/4·16-s − 0.849·17-s − 1.97·18-s − 0.310·19-s + 0.330·20-s − 1.54·21-s + 0.213·22-s + 0.393·23-s − 0.687·24-s − 0.561·25-s − 0.983·26-s + 3.47·27-s − 0.397·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.266671553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266671553\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 11 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 - 91 T + p^{7} T^{2} \) |
| 5 | \( 1 - 37 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 722 T + p^{7} T^{2} \) |
| 13 | \( 1 - 11020 T + p^{7} T^{2} \) |
| 17 | \( 1 + 17210 T + p^{7} T^{2} \) |
| 19 | \( 1 + 9288 T + p^{7} T^{2} \) |
| 23 | \( 1 - 22971 T + p^{7} T^{2} \) |
| 29 | \( 1 - 134272 T + p^{7} T^{2} \) |
| 31 | \( 1 + 287765 T + p^{7} T^{2} \) |
| 37 | \( 1 + 316397 T + p^{7} T^{2} \) |
| 41 | \( 1 + 335968 T + p^{7} T^{2} \) |
| 43 | \( 1 + 858110 T + p^{7} T^{2} \) |
| 47 | \( 1 - 587680 T + p^{7} T^{2} \) |
| 53 | \( 1 + 244238 T + p^{7} T^{2} \) |
| 59 | \( 1 + 163287 T + p^{7} T^{2} \) |
| 61 | \( 1 - 37660 p T + p^{7} T^{2} \) |
| 67 | \( 1 + 3428283 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1542953 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2216316 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1526014 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1650370 T + p^{7} T^{2} \) |
| 89 | \( 1 - 5760847 T + p^{7} T^{2} \) |
| 97 | \( 1 + 5750759 T + p^{7} T^{2} \) |
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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
−16.13025491106890823577394646999, −15.23411323583171878874349366748, −13.76717319239749227456347896013, −13.00956206096740076900201369833, −10.40082305189310242155641163440, −9.255310515619309659573404786073, −8.406191877583896639070274387489, −6.76548516524270890449383429377, −3.41956656198945190330483516321, −1.88627257680106004495870978430,
1.88627257680106004495870978430, 3.41956656198945190330483516321, 6.76548516524270890449383429377, 8.406191877583896639070274387489, 9.255310515619309659573404786073, 10.40082305189310242155641163440, 13.00956206096740076900201369833, 13.76717319239749227456347896013, 15.23411323583171878874349366748, 16.13025491106890823577394646999
