| L(s) = 1 | − 2·5-s + 4·7-s − 3·9-s + 2·11-s + 13-s − 17-s + 2·23-s − 25-s − 8·29-s − 8·31-s − 8·35-s + 6·37-s + 12·41-s − 4·43-s + 6·45-s − 8·47-s + 9·49-s + 6·53-s − 4·55-s + 4·59-s + 8·61-s − 12·63-s − 2·65-s + 8·67-s − 8·71-s + 8·73-s + 8·77-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 9-s + 0.603·11-s + 0.277·13-s − 0.242·17-s + 0.417·23-s − 1/5·25-s − 1.48·29-s − 1.43·31-s − 1.35·35-s + 0.986·37-s + 1.87·41-s − 0.609·43-s + 0.894·45-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s + 0.520·59-s + 1.02·61-s − 1.51·63-s − 0.248·65-s + 0.977·67-s − 0.949·71-s + 0.936·73-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.763705263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.763705263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.29920596360173, −15.43202864312505, −14.78467335676638, −14.65678680680600, −14.15353241600001, −13.30264893323299, −12.78570630140377, −11.93807150500177, −11.47752404239153, −11.12182482154386, −10.90548867232979, −9.692543677663862, −9.126915915565828, −8.473059708356226, −8.081327629699036, −7.483044435043449, −6.915940769338393, −5.852419584525023, −5.480679252851308, −4.640385553306849, −4.008746565353977, −3.455670505032799, −2.384263736526928, −1.646933021013913, −0.5936069925683396,
0.5936069925683396, 1.646933021013913, 2.384263736526928, 3.455670505032799, 4.008746565353977, 4.640385553306849, 5.480679252851308, 5.852419584525023, 6.915940769338393, 7.483044435043449, 8.081327629699036, 8.473059708356226, 9.126915915565828, 9.692543677663862, 10.90548867232979, 11.12182482154386, 11.47752404239153, 11.93807150500177, 12.78570630140377, 13.30264893323299, 14.15353241600001, 14.65678680680600, 14.78467335676638, 15.43202864312505, 16.29920596360173
