| L(s) = 1 | − 5-s − 2·7-s + 3·11-s + 13-s − 8·19-s + 6·23-s − 4·25-s + 6·29-s − 4·31-s + 2·35-s + 41-s − 7·43-s − 3·49-s − 2·53-s − 3·55-s + 4·59-s + 13·61-s − 65-s − 12·67-s + 4·73-s − 6·77-s + 9·79-s + 9·83-s + 89-s − 2·91-s + 8·95-s + 16·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.904·11-s + 0.277·13-s − 1.83·19-s + 1.25·23-s − 4/5·25-s + 1.11·29-s − 0.718·31-s + 0.338·35-s + 0.156·41-s − 1.06·43-s − 3/7·49-s − 0.274·53-s − 0.404·55-s + 0.520·59-s + 1.66·61-s − 0.124·65-s − 1.46·67-s + 0.468·73-s − 0.683·77-s + 1.01·79-s + 0.987·83-s + 0.105·89-s − 0.209·91-s + 0.820·95-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.93692030144968, −12.68283807659909, −11.98938405591710, −11.73875428618710, −11.15053016786767, −10.76729779423252, −10.23273236803959, −9.828106613675302, −9.207434664642562, −8.825681574841088, −8.493375059020863, −7.887569702564939, −7.365026199542802, −6.733459139341576, −6.369675984865314, −6.235771774208231, −5.280545378314804, −4.882924929309636, −4.188111063402073, −3.802169697379908, −3.387031230888873, −2.707635690840475, −2.098531684604614, −1.440239183076518, −0.6913970068234650, 0,
0.6913970068234650, 1.440239183076518, 2.098531684604614, 2.707635690840475, 3.387031230888873, 3.802169697379908, 4.188111063402073, 4.882924929309636, 5.280545378314804, 6.235771774208231, 6.369675984865314, 6.733459139341576, 7.365026199542802, 7.887569702564939, 8.493375059020863, 8.825681574841088, 9.207434664642562, 9.828106613675302, 10.23273236803959, 10.76729779423252, 11.15053016786767, 11.73875428618710, 11.98938405591710, 12.68283807659909, 12.93692030144968
