| L(s) = 1 | + 6.05·2-s − 91.3·4-s + 125·5-s − 1.43e3·7-s − 1.32e3·8-s + 756.·10-s − 5.81e3·11-s − 6.32e3·13-s − 8.66e3·14-s + 3.65e3·16-s − 2.90e4·17-s − 4.37e4·19-s − 1.14e4·20-s − 3.52e4·22-s + 5.52e3·23-s + 1.56e4·25-s − 3.83e4·26-s + 1.30e5·28-s + 1.50e5·29-s − 2.05e5·31-s + 1.92e5·32-s − 1.75e5·34-s − 1.78e5·35-s − 4.53e5·37-s − 2.64e5·38-s − 1.65e5·40-s + 8.10e4·41-s + ⋯ |
| L(s) = 1 | + 0.535·2-s − 0.713·4-s + 0.447·5-s − 1.57·7-s − 0.916·8-s + 0.239·10-s − 1.31·11-s − 0.798·13-s − 0.843·14-s + 0.222·16-s − 1.43·17-s − 1.46·19-s − 0.319·20-s − 0.705·22-s + 0.0946·23-s + 0.199·25-s − 0.427·26-s + 1.12·28-s + 1.14·29-s − 1.23·31-s + 1.03·32-s − 0.766·34-s − 0.705·35-s − 1.47·37-s − 0.782·38-s − 0.410·40-s + 0.183·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.005161845258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005161845258\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| good | 2 | \( 1 - 6.05T + 128T^{2} \) |
| 7 | \( 1 + 1.43e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.81e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.32e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.90e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.37e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.52e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.50e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.05e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.10e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.95e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.83e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.35e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.17e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.34e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.20e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.00e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.10e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.98e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.24e7T + 8.07e13T^{2} \) |
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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
−10.07971740040209651668443552117, −9.195635705072095516018692014165, −8.483979307632872807253125696274, −7.00637500975893039940815470016, −6.20923880862527767011365714974, −5.23450658107991836421478387319, −4.31494066478683495702698776978, −3.12345496353379609037947907982, −2.29078276841728089148689136265, −0.02793809036076266733551155099,
0.02793809036076266733551155099, 2.29078276841728089148689136265, 3.12345496353379609037947907982, 4.31494066478683495702698776978, 5.23450658107991836421478387319, 6.20923880862527767011365714974, 7.00637500975893039940815470016, 8.483979307632872807253125696274, 9.195635705072095516018692014165, 10.07971740040209651668443552117
