| L(s) = 1 | − 5·11-s + 2·13-s + 6·17-s − 2·19-s + 5·23-s + 5·29-s − 4·31-s + 37-s − 12·41-s − 5·43-s − 2·47-s − 14·53-s − 2·59-s + 5·67-s − 9·71-s − 10·73-s − 11·79-s − 16·83-s − 14·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.50·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.04·23-s + 0.928·29-s − 0.718·31-s + 0.164·37-s − 1.87·41-s − 0.762·43-s − 0.291·47-s − 1.92·53-s − 0.260·59-s + 0.610·67-s − 1.06·71-s − 1.17·73-s − 1.23·79-s − 1.75·83-s − 1.48·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.035819684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035819684\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.13067735982148, −12.82899453595278, −12.28838215645491, −11.78816916765878, −11.24481470264005, −10.75239537437430, −10.38672549999465, −9.902695079765183, −9.526524229244511, −8.721624794558620, −8.286276393667378, −8.083486375286296, −7.333562830848594, −6.990192528444545, −6.359783251119721, −5.697716165160943, −5.371353421620496, −4.849295555261091, −4.322645641107934, −3.498164891311076, −2.986255942727593, −2.760001411503829, −1.638817756789792, −1.360684550643801, −0.2890463075868126,
0.2890463075868126, 1.360684550643801, 1.638817756789792, 2.760001411503829, 2.986255942727593, 3.498164891311076, 4.322645641107934, 4.849295555261091, 5.371353421620496, 5.697716165160943, 6.359783251119721, 6.990192528444545, 7.333562830848594, 8.083486375286296, 8.286276393667378, 8.721624794558620, 9.526524229244511, 9.902695079765183, 10.38672549999465, 10.75239537437430, 11.24481470264005, 11.78816916765878, 12.28838215645491, 12.82899453595278, 13.13067735982148
