| L(s) = 1 | + 28.4·3-s + 97.1·5-s − 3.39·7-s + 566.·9-s − 121·11-s − 655.·13-s + 2.76e3·15-s − 1.67e3·17-s − 1.19e3·19-s − 96.5·21-s − 1.99e3·23-s + 6.31e3·25-s + 9.20e3·27-s − 291.·29-s − 715.·31-s − 3.44e3·33-s − 329.·35-s + 5.99e3·37-s − 1.86e4·39-s + 1.76e4·41-s + 2.23e4·43-s + 5.50e4·45-s − 2.65e4·47-s − 1.67e4·49-s − 4.75e4·51-s + 1.29e4·53-s − 1.17e4·55-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 1.73·5-s − 0.0261·7-s + 2.33·9-s − 0.301·11-s − 1.07·13-s + 3.17·15-s − 1.40·17-s − 0.759·19-s − 0.0477·21-s − 0.785·23-s + 2.02·25-s + 2.43·27-s − 0.0642·29-s − 0.133·31-s − 0.550·33-s − 0.0454·35-s + 0.720·37-s − 1.96·39-s + 1.63·41-s + 1.84·43-s + 4.05·45-s − 1.75·47-s − 0.999·49-s − 2.56·51-s + 0.633·53-s − 0.524·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.989694623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.989694623\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 - 28.4T + 243T^{2} \) |
| 5 | \( 1 - 97.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 3.39T + 1.68e4T^{2} \) |
| 13 | \( 1 + 655.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.67e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.19e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.99e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 291.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 715.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.76e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.23e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.65e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.29e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.75e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.42e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.81e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.09e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.61e5T + 8.58e9T^{2} \) |
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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
−13.31885986283792997653345059887, −12.80212620243840956935283447629, −10.53285693550137748266447492902, −9.564149964841720487095871394431, −9.003624094073022383651763603349, −7.69265958865418670348219735596, −6.32927869559475797665140040646, −4.52864184252007483940372753774, −2.61890124436495795046838131686, −1.97604735869199125757878149704,
1.97604735869199125757878149704, 2.61890124436495795046838131686, 4.52864184252007483940372753774, 6.32927869559475797665140040646, 7.69265958865418670348219735596, 9.003624094073022383651763603349, 9.564149964841720487095871394431, 10.53285693550137748266447492902, 12.80212620243840956935283447629, 13.31885986283792997653345059887
