| L(s) = 1 | + 5-s − 4·7-s − 11-s + 2·13-s − 2·17-s − 4·19-s − 4·23-s + 25-s − 6·29-s − 4·35-s − 10·37-s + 6·41-s + 12·43-s − 4·47-s + 9·49-s + 6·53-s − 55-s − 4·59-s + 10·61-s + 2·65-s + 12·67-s − 4·71-s + 10·73-s + 4·77-s − 4·79-s + 4·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.676·35-s − 1.64·37-s + 0.937·41-s + 1.82·43-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s + 1.46·67-s − 0.474·71-s + 1.17·73-s + 0.455·77-s − 0.450·79-s + 0.439·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.198604374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198604374\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−7.78965359455486421637707631628, −7.01299420411563816546019977542, −6.38126558368074953208304890214, −5.92957662563690492395751345040, −5.19993351252464527374411354888, −4.05385861023260388282179375127, −3.61940733835439603523436833531, −2.61424817284399953231027310483, −1.94604140615759817155300036227, −0.51584605831208707950917523839,
0.51584605831208707950917523839, 1.94604140615759817155300036227, 2.61424817284399953231027310483, 3.61940733835439603523436833531, 4.05385861023260388282179375127, 5.19993351252464527374411354888, 5.92957662563690492395751345040, 6.38126558368074953208304890214, 7.01299420411563816546019977542, 7.78965359455486421637707631628
