| L(s) = 1 | − 7-s − 4·11-s − 6·13-s + 2·17-s − 6·19-s + 2·23-s + 6·29-s − 2·31-s − 4·37-s − 8·41-s − 4·43-s + 4·47-s + 49-s − 6·53-s + 4·59-s − 14·61-s + 4·67-s + 10·73-s + 4·77-s + 16·83-s − 8·89-s + 6·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s − 1.66·13-s + 0.485·17-s − 1.37·19-s + 0.417·23-s + 1.11·29-s − 0.359·31-s − 0.657·37-s − 1.24·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s − 1.79·61-s + 0.488·67-s + 1.17·73-s + 0.455·77-s + 1.75·83-s − 0.847·89-s + 0.628·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.94030638462730, −13.52661844969192, −12.88239099264920, −12.50837489963830, −12.22725410159005, −11.65243520106947, −10.83994974871802, −10.55752029313493, −10.05981722175058, −9.694912327193982, −9.041989103772740, −8.463517790128067, −7.980560234913148, −7.525163853268826, −6.856010893077424, −6.586318163590478, −5.765000559938649, −5.235668061334095, −4.767305123426919, −4.344272435401614, −3.320165426495213, −3.023251077314018, −2.249702796718234, −1.864348407663698, −0.6401256996200569, 0,
0.6401256996200569, 1.864348407663698, 2.249702796718234, 3.023251077314018, 3.320165426495213, 4.344272435401614, 4.767305123426919, 5.235668061334095, 5.765000559938649, 6.586318163590478, 6.856010893077424, 7.525163853268826, 7.980560234913148, 8.463517790128067, 9.041989103772740, 9.694912327193982, 10.05981722175058, 10.55752029313493, 10.83994974871802, 11.65243520106947, 12.22725410159005, 12.50837489963830, 12.88239099264920, 13.52661844969192, 13.94030638462730
