| L(s) = 1 | + 5-s − 3·7-s + 5·13-s + 4·17-s + 7·19-s − 4·23-s + 25-s − 3·29-s − 3·31-s − 3·35-s − 4·37-s + 11·41-s − 43-s − 2·47-s + 2·49-s − 10·53-s − 4·59-s − 11·61-s + 5·65-s − 67-s − 14·71-s − 3·73-s − 13·79-s − 8·83-s + 4·85-s + 4·89-s − 15·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 1.38·13-s + 0.970·17-s + 1.60·19-s − 0.834·23-s + 1/5·25-s − 0.557·29-s − 0.538·31-s − 0.507·35-s − 0.657·37-s + 1.71·41-s − 0.152·43-s − 0.291·47-s + 2/7·49-s − 1.37·53-s − 0.520·59-s − 1.40·61-s + 0.620·65-s − 0.122·67-s − 1.66·71-s − 0.351·73-s − 1.46·79-s − 0.878·83-s + 0.433·85-s + 0.423·89-s − 1.57·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.65563529716636, −14.71820539928510, −14.14474849738328, −13.85336074551572, −13.21443808824016, −12.79862004444034, −12.28569803842502, −11.61092235790150, −11.13971486227855, −10.39469783991609, −10.00094421853987, −9.375232338975099, −9.098681651617793, −8.323203055118812, −7.549001963917171, −7.270599911349055, −6.240725330754514, −6.007934037280167, −5.551335664782913, −4.673198554280015, −3.806357716412780, −3.288369884721706, −2.873196736981550, −1.686953212720659, −1.137364769803711, 0,
1.137364769803711, 1.686953212720659, 2.873196736981550, 3.288369884721706, 3.806357716412780, 4.673198554280015, 5.551335664782913, 6.007934037280167, 6.240725330754514, 7.270599911349055, 7.549001963917171, 8.323203055118812, 9.098681651617793, 9.375232338975099, 10.00094421853987, 10.39469783991609, 11.13971486227855, 11.61092235790150, 12.28569803842502, 12.79862004444034, 13.21443808824016, 13.85336074551572, 14.14474849738328, 14.71820539928510, 15.65563529716636
