| L(s) = 1 | − 0.732·5-s − 7-s − 3.46·11-s + 4·13-s + 3.26·17-s + 19-s − 0.535·23-s − 4.46·25-s − 2.73·29-s − 3.46·31-s + 0.732·35-s − 7.46·37-s + 11.4·41-s − 8.92·43-s + 5.66·47-s + 49-s + 8.19·53-s + 2.53·55-s + 8·59-s + 4.53·61-s − 2.92·65-s − 4·67-s − 6.19·71-s + 0.535·73-s + 3.46·77-s − 5.46·79-s − 0.196·83-s + ⋯ |
| L(s) = 1 | − 0.327·5-s − 0.377·7-s − 1.04·11-s + 1.10·13-s + 0.792·17-s + 0.229·19-s − 0.111·23-s − 0.892·25-s − 0.507·29-s − 0.622·31-s + 0.123·35-s − 1.22·37-s + 1.79·41-s − 1.36·43-s + 0.825·47-s + 0.142·49-s + 1.12·53-s + 0.341·55-s + 1.04·59-s + 0.580·61-s − 0.363·65-s − 0.488·67-s − 0.735·71-s + 0.0627·73-s + 0.394·77-s − 0.614·79-s − 0.0215·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.466552494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.466552494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 0.732T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 23 | \( 1 + 0.535T + 23T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 - 5.66T + 47T^{2} \) |
| 53 | \( 1 - 8.19T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 4.53T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 + 0.196T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.48966471687452736005384545564, −7.30300186993056739819950634438, −6.16354047907991182134082748996, −5.69475418493493397485105148642, −5.06474179561794531607418513894, −3.96967273762296694840005577259, −3.55163173685440234926923557660, −2.68504311789875791285292515414, −1.71977579391393835244646285291, −0.57458679088412248307946885761,
0.57458679088412248307946885761, 1.71977579391393835244646285291, 2.68504311789875791285292515414, 3.55163173685440234926923557660, 3.96967273762296694840005577259, 5.06474179561794531607418513894, 5.69475418493493397485105148642, 6.16354047907991182134082748996, 7.30300186993056739819950634438, 7.48966471687452736005384545564
