| L(s) = 1 | − 3.76·7-s − 3.54·11-s + 1.00·13-s + 1.56·17-s + 7.21·19-s + 6.18·23-s − 3·29-s − 5.78·31-s + 0.851·37-s − 3.11·41-s − 2.70·43-s + 9.74·47-s + 7.21·49-s − 11.2·53-s − 9.66·59-s − 8.21·61-s − 2.91·67-s + 7.21·71-s + 16.7·73-s + 13.3·77-s + 10.9·79-s + 10.1·83-s − 5.33·89-s − 3.78·91-s − 3.86·97-s + 4.75·101-s + 18.0·103-s + ⋯ |
| L(s) = 1 | − 1.42·7-s − 1.06·11-s + 0.278·13-s + 0.378·17-s + 1.65·19-s + 1.28·23-s − 0.557·29-s − 1.03·31-s + 0.139·37-s − 0.487·41-s − 0.412·43-s + 1.42·47-s + 1.03·49-s − 1.54·53-s − 1.25·59-s − 1.05·61-s − 0.356·67-s + 0.855·71-s + 1.96·73-s + 1.52·77-s + 1.23·79-s + 1.11·83-s − 0.565·89-s − 0.396·91-s − 0.392·97-s + 0.473·101-s + 1.78·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.76T + 7T^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 - 6.18T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 - 0.851T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 + 2.70T + 43T^{2} \) |
| 47 | \( 1 - 9.74T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 9.66T + 59T^{2} \) |
| 61 | \( 1 + 8.21T + 61T^{2} \) |
| 67 | \( 1 + 2.91T + 67T^{2} \) |
| 71 | \( 1 - 7.21T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 5.33T + 89T^{2} \) |
| 97 | \( 1 + 3.86T + 97T^{2} \) |
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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
−7.50933937438454119500787402913, −6.82149606421810486220538412295, −6.08887977238123371605138816190, −5.38994217981777278687292104525, −4.85638375804541112361058322895, −3.47410300903029868347481691643, −3.32316051333486337095946712437, −2.40999332581044834396919392021, −1.12073339115586318843276750027, 0,
1.12073339115586318843276750027, 2.40999332581044834396919392021, 3.32316051333486337095946712437, 3.47410300903029868347481691643, 4.85638375804541112361058322895, 5.38994217981777278687292104525, 6.08887977238123371605138816190, 6.82149606421810486220538412295, 7.50933937438454119500787402913
