| L(s) = 1 | + 4·5-s − 2·7-s − 5·11-s + 2·13-s + 3·17-s − 19-s + 6·23-s + 11·25-s − 2·29-s − 4·31-s − 8·35-s + 8·37-s − 41-s + 7·43-s − 2·47-s − 3·49-s − 4·53-s − 20·55-s + 5·59-s + 8·65-s + 13·67-s − 8·71-s + 3·73-s + 10·77-s + 8·79-s + 12·83-s + 12·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s − 1.50·11-s + 0.554·13-s + 0.727·17-s − 0.229·19-s + 1.25·23-s + 11/5·25-s − 0.371·29-s − 0.718·31-s − 1.35·35-s + 1.31·37-s − 0.156·41-s + 1.06·43-s − 0.291·47-s − 3/7·49-s − 0.549·53-s − 2.69·55-s + 0.650·59-s + 0.992·65-s + 1.58·67-s − 0.949·71-s + 0.351·73-s + 1.13·77-s + 0.900·79-s + 1.31·83-s + 1.30·85-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
| \(L(1)\) |
\(\approx\) |
\(2.503801346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.503801346\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
| 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
−8.240856915758520522356088138500, −7.44712327425528078975860928297, −6.59974455735529645673910898610, −5.98116686810622067066538802753, −5.43646268942204203157631928118, −4.82921977656014299866731136572, −3.45785097536163636692546343378, −2.73893739226669684936996532796, −2.03206906194722746237995457655, −0.861498677976581208596394326796,
0.861498677976581208596394326796, 2.03206906194722746237995457655, 2.73893739226669684936996532796, 3.45785097536163636692546343378, 4.82921977656014299866731136572, 5.43646268942204203157631928118, 5.98116686810622067066538802753, 6.59974455735529645673910898610, 7.44712327425528078975860928297, 8.240856915758520522356088138500
