| L(s) = 1 | + 1.05·3-s − 5-s + 3.74·7-s − 1.88·9-s − 4.91·11-s − 1.25·13-s − 1.05·15-s + 3.28·17-s + 2.70·19-s + 3.95·21-s + 0.0904·23-s + 25-s − 5.16·27-s − 3.78·29-s − 2.75·31-s − 5.19·33-s − 3.74·35-s + 6.68·37-s − 1.32·39-s − 8.86·41-s + 9.41·43-s + 1.88·45-s − 4.56·47-s + 7.02·49-s + 3.46·51-s − 12.5·53-s + 4.91·55-s + ⋯ |
| L(s) = 1 | + 0.610·3-s − 0.447·5-s + 1.41·7-s − 0.627·9-s − 1.48·11-s − 0.346·13-s − 0.272·15-s + 0.796·17-s + 0.621·19-s + 0.863·21-s + 0.0188·23-s + 0.200·25-s − 0.993·27-s − 0.703·29-s − 0.494·31-s − 0.904·33-s − 0.632·35-s + 1.09·37-s − 0.211·39-s − 1.38·41-s + 1.43·43-s + 0.280·45-s − 0.665·47-s + 1.00·49-s + 0.485·51-s − 1.71·53-s + 0.663·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 1.05T + 3T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + 4.91T + 11T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 - 0.0904T + 23T^{2} \) |
| 29 | \( 1 + 3.78T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 - 6.68T + 37T^{2} \) |
| 41 | \( 1 + 8.86T + 41T^{2} \) |
| 43 | \( 1 - 9.41T + 43T^{2} \) |
| 47 | \( 1 + 4.56T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 6.26T + 59T^{2} \) |
| 61 | \( 1 - 5.05T + 61T^{2} \) |
| 67 | \( 1 - 2.00T + 67T^{2} \) |
| 71 | \( 1 + 3.08T + 71T^{2} \) |
| 73 | \( 1 - 1.79T + 73T^{2} \) |
| 79 | \( 1 + 6.31T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 6.23T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−7.68373582642794267743427508996, −7.19125540370754860144078884605, −5.85237851972613626031477898221, −5.30737618310498804879540280099, −4.79055326471501531013496851603, −3.83226849723142827006000868211, −2.98603528430651123072535058592, −2.36980631600260063114855047665, −1.38332726423575101759535163903, 0,
1.38332726423575101759535163903, 2.36980631600260063114855047665, 2.98603528430651123072535058592, 3.83226849723142827006000868211, 4.79055326471501531013496851603, 5.30737618310498804879540280099, 5.85237851972613626031477898221, 7.19125540370754860144078884605, 7.68373582642794267743427508996
