L(s) = 1 | − 5-s − 3.44·7-s − 1.44·11-s − 5.89·13-s − 4.89·17-s + 4·19-s − 5.44·23-s − 4·25-s + 0.101·29-s − 2.55·31-s + 3.44·35-s + 0.898·37-s + 11.8·41-s − 2.34·43-s − 6.34·47-s + 4.89·49-s + 8.89·53-s + 1.44·55-s − 14.3·59-s + 7.89·61-s + 5.89·65-s + 12.3·67-s + 7.79·71-s − 4.89·73-s + 5·77-s − 13.4·79-s + 0.550·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.30·7-s − 0.437·11-s − 1.63·13-s − 1.18·17-s + 0.917·19-s − 1.13·23-s − 0.800·25-s + 0.0187·29-s − 0.458·31-s + 0.583·35-s + 0.147·37-s + 1.85·41-s − 0.358·43-s − 0.926·47-s + 0.699·49-s + 1.22·53-s + 0.195·55-s − 1.86·59-s + 1.01·61-s + 0.731·65-s + 1.50·67-s + 0.925·71-s − 0.573·73-s + 0.569·77-s − 1.51·79-s + 0.0604·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5131296979\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5131296979\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 + 5.89T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 5.44T + 23T^{2} \) |
| 29 | \( 1 - 0.101T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 - 0.898T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 + 6.34T + 47T^{2} \) |
| 53 | \( 1 - 8.89T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 7.89T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 7.79T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 0.550T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 3.89T + 97T^{2} \) |
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
−8.030200922843012301921071779697, −7.48692935031461603583495669770, −6.85824021793040372397054799092, −6.10017260962653808080682863647, −5.32765191197100769243608382505, −4.44861806040112754251153922618, −3.72049838356548262665664224823, −2.80479861760478796836196446731, −2.13554095068885957293860379543, −0.35876913074410584393685452131,
0.35876913074410584393685452131, 2.13554095068885957293860379543, 2.80479861760478796836196446731, 3.72049838356548262665664224823, 4.44861806040112754251153922618, 5.32765191197100769243608382505, 6.10017260962653808080682863647, 6.85824021793040372397054799092, 7.48692935031461603583495669770, 8.030200922843012301921071779697