| L(s) = 1 | + 7-s − 2·13-s − 2·19-s − 6·29-s − 8·31-s + 4·37-s − 6·41-s + 2·43-s + 6·47-s + 49-s + 6·53-s + 12·59-s + 8·61-s + 2·67-s + 6·71-s − 2·73-s + 16·79-s − 6·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.554·13-s − 0.458·19-s − 1.11·29-s − 1.43·31-s + 0.657·37-s − 0.937·41-s + 0.304·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s + 1.56·59-s + 1.02·61-s + 0.244·67-s + 0.712·71-s − 0.234·73-s + 1.80·79-s − 0.635·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.50401577020100, −15.03443301967451, −14.59354266272997, −14.15349847043732, −13.39930071386726, −12.95461770275739, −12.45710161423636, −11.80857718855783, −11.27318419989347, −10.83712430105197, −10.13418607263805, −9.668468837249962, −8.937873008694224, −8.575031251331864, −7.727839466939940, −7.364907963935085, −6.727240953290798, −5.987946462421398, −5.317886768800363, −4.915805961990843, −3.918610012592504, −3.652072055111557, −2.448374292213280, −2.083567389635565, −1.045020633457078, 0,
1.045020633457078, 2.083567389635565, 2.448374292213280, 3.652072055111557, 3.918610012592504, 4.915805961990843, 5.317886768800363, 5.987946462421398, 6.727240953290798, 7.364907963935085, 7.727839466939940, 8.575031251331864, 8.937873008694224, 9.668468837249962, 10.13418607263805, 10.83712430105197, 11.27318419989347, 11.80857718855783, 12.45710161423636, 12.95461770275739, 13.39930071386726, 14.15349847043732, 14.59354266272997, 15.03443301967451, 15.50401577020100
