| L(s) = 1 | − 5-s + 2·13-s + 2·19-s + 25-s − 6·29-s − 8·31-s + 4·37-s − 6·41-s − 2·43-s + 6·47-s − 6·53-s + 12·59-s + 8·61-s − 2·65-s − 2·67-s + 6·71-s − 2·73-s − 16·79-s − 6·89-s − 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s + 0.458·19-s + 1/5·25-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 − 0.824·53-s + 1.56·59-s + 1.02·61-s − 0.248·65-s − 0.244·67-s + 0.712·71-s − 0.234·73-s − 1.80·79-s − 0.635·89-s − 0.205·95-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593830109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593830109\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.16405007511072, −12.98374298485227, −12.60679257343997, −11.77991491104583, −11.45616851149223, −11.21196608734839, −10.49499735238248, −10.09167199279384, −9.519337097129741, −8.952302995251069, −8.632105760356928, −8.000114237292844, −7.483159927561760, −7.122986827123040, −6.530232823658615, −5.881417487710071, −5.436298374074009, −4.938751020366509, −4.176692312946480, −3.719249913135954, −3.316390098477985, −2.532021570500060, −1.867611311544058, −1.212068827650180, −0.3937936702584796,
0.3937936702584796, 1.212068827650180, 1.867611311544058, 2.532021570500060, 3.316390098477985, 3.719249913135954, 4.176692312946480, 4.938751020366509, 5.436298374074009, 5.881417487710071, 6.530232823658615, 7.122986827123040, 7.483159927561760, 8.000114237292844, 8.632105760356928, 8.952302995251069, 9.519337097129741, 10.09167199279384, 10.49499735238248, 11.21196608734839, 11.45616851149223, 11.77991491104583, 12.60679257343997, 12.98374298485227, 13.16405007511072
