| L(s) = 1 | + 2·5-s + 4.24·7-s + 1.41·11-s − 4·13-s + 17-s + 2.82·19-s + 4.24·23-s − 25-s + 6·29-s − 7.07·31-s + 8.48·35-s − 2·37-s + 6·41-s − 8.48·43-s + 11.3·47-s + 10.9·49-s + 6·53-s + 2.82·55-s − 8.48·59-s + 6·61-s − 8·65-s + 5.65·67-s − 7.07·71-s + 10·73-s + 6·77-s + 12.7·79-s − 14.1·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.60·7-s + 0.426·11-s − 1.10·13-s + 0.242·17-s + 0.648·19-s + 0.884·23-s − 0.200·25-s + 1.11·29-s − 1.27·31-s + 1.43·35-s − 0.328·37-s + 0.937·41-s − 1.29·43-s + 1.65·47-s + 1.57·49-s + 0.824·53-s + 0.381·55-s − 1.10·59-s + 0.768·61-s − 0.992·65-s + 0.691·67-s − 0.839·71-s + 1.17·73-s + 0.683·77-s + 1.43·79-s − 1.55·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.040712065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.040712065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
| 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
−8.239900779541384400026437055018, −7.50447969924708564109505258679, −6.97816031607951331509385936280, −5.93939012329785636914119764507, −5.19785969172403231592485555678, −4.84335814291689097956092733081, −3.82483693447345354328771517770, −2.62349429029595045490590333967, −1.89776442955403834689420296428, −1.03214523144953008050922929443,
1.03214523144953008050922929443, 1.89776442955403834689420296428, 2.62349429029595045490590333967, 3.82483693447345354328771517770, 4.84335814291689097956092733081, 5.19785969172403231592485555678, 5.93939012329785636914119764507, 6.97816031607951331509385936280, 7.50447969924708564109505258679, 8.239900779541384400026437055018
