| L(s) = 1 | + 2·7-s − 3·9-s − 4·11-s − 2·13-s − 4·17-s + 19-s − 6·23-s + 6·29-s + 4·31-s − 10·37-s − 10·41-s − 2·43-s − 6·47-s − 3·49-s + 10·53-s − 2·61-s − 6·63-s − 8·67-s − 4·71-s − 4·73-s − 8·77-s − 4·79-s + 9·81-s + 18·83-s − 2·89-s − 4·91-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s − 1.20·11-s − 0.554·13-s − 0.970·17-s + 0.229·19-s − 1.25·23-s + 1.11·29-s + 0.718·31-s − 1.64·37-s − 1.56·41-s − 0.304·43-s − 0.875·47-s − 3/7·49-s + 1.37·53-s − 0.256·61-s − 0.755·63-s − 0.977·67-s − 0.474·71-s − 0.468·73-s − 0.911·77-s − 0.450·79-s + 81-s + 1.97·83-s − 0.211·89-s − 0.419·91-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6784211342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6784211342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.10788360766588, −14.63591040162414, −13.88990011440446, −13.72705788477801, −13.13744925189553, −12.30442995308104, −11.84691704230534, −11.54303608190131, −10.79340721770672, −10.20063245167792, −10.04139016081147, −8.853056238810216, −8.658789712730202, −8.003228766947811, −7.641573015718220, −6.765778862114034, −6.276877088738288, −5.450457045467083, −5.007806169997759, −4.579193527580036, −3.607130834964076, −2.874743259551771, −2.295496907753385, −1.622291161487484, −0.2968376346866832,
0.2968376346866832, 1.622291161487484, 2.295496907753385, 2.874743259551771, 3.607130834964076, 4.579193527580036, 5.007806169997759, 5.450457045467083, 6.276877088738288, 6.765778862114034, 7.641573015718220, 8.003228766947811, 8.658789712730202, 8.853056238810216, 10.04139016081147, 10.20063245167792, 10.79340721770672, 11.54303608190131, 11.84691704230534, 12.30442995308104, 13.13744925189553, 13.72705788477801, 13.88990011440446, 14.63591040162414, 15.10788360766588
