| L(s) = 1 | − 0.618·3-s − 1.61·7-s − 2.61·9-s + 11-s − 2.23·13-s − 1.85·17-s − 3.47·19-s + 1.00·21-s + 9.32·23-s + 3.47·27-s − 6.61·29-s − 0.236·31-s − 0.618·33-s + 6.23·37-s + 1.38·39-s − 9.94·41-s − 0.472·43-s − 8.70·47-s − 4.38·49-s + 1.14·51-s + 2.85·53-s + 2.14·57-s + 8.23·59-s − 4.09·61-s + 4.23·63-s − 5.76·69-s + 13.7·71-s + ⋯ |
| L(s) = 1 | − 0.356·3-s − 0.611·7-s − 0.872·9-s + 0.301·11-s − 0.620·13-s − 0.449·17-s − 0.796·19-s + 0.218·21-s + 1.94·23-s + 0.668·27-s − 1.22·29-s − 0.0423·31-s − 0.107·33-s + 1.02·37-s + 0.221·39-s − 1.55·41-s − 0.0720·43-s − 1.27·47-s − 0.625·49-s + 0.160·51-s + 0.392·53-s + 0.284·57-s + 1.07·59-s − 0.523·61-s + 0.533·63-s − 0.693·69-s + 1.62·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9741402730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9741402730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 - 9.32T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 + 0.236T + 31T^{2} \) |
| 37 | \( 1 - 6.23T + 37T^{2} \) |
| 41 | \( 1 + 9.94T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 + 8.70T + 47T^{2} \) |
| 53 | \( 1 - 2.85T + 53T^{2} \) |
| 59 | \( 1 - 8.23T + 59T^{2} \) |
| 61 | \( 1 + 4.09T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 4.09T + 73T^{2} \) |
| 79 | \( 1 - 3.85T + 79T^{2} \) |
| 83 | \( 1 - 7.85T + 83T^{2} \) |
| 89 | \( 1 - 7.56T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.456288159585945761908661817888, −7.58645194732335197381003657582, −6.65707079394906933613438055662, −6.38838191708988487489082413256, −5.30244395739414736816967678476, −4.84251279691540244497751307509, −3.69086167182309022066126049034, −2.96160042721606733352207289286, −2.01455871889353281279335701677, −0.54008461103459087441154626544,
0.54008461103459087441154626544, 2.01455871889353281279335701677, 2.96160042721606733352207289286, 3.69086167182309022066126049034, 4.84251279691540244497751307509, 5.30244395739414736816967678476, 6.38838191708988487489082413256, 6.65707079394906933613438055662, 7.58645194732335197381003657582, 8.456288159585945761908661817888
