| L(s) = 1 | + 2·7-s − 6·11-s + 4·13-s + 6·17-s + 4·19-s − 6·29-s + 4·31-s − 8·37-s + 8·43-s − 3·49-s + 6·53-s − 6·59-s + 2·61-s − 4·67-s + 12·71-s + 10·73-s − 12·77-s + 4·79-s + 12·83-s + 12·89-s + 8·91-s − 2·97-s − 6·101-s + 2·103-s + 12·107-s + 2·109-s − 6·113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.80·11-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 1.11·29-s + 0.718·31-s − 1.31·37-s + 1.21·43-s − 3/7·49-s + 0.824·53-s − 0.781·59-s + 0.256·61-s − 0.488·67-s + 1.42·71-s + 1.17·73-s − 1.36·77-s + 0.450·79-s + 1.31·83-s + 1.27·89-s + 0.838·91-s − 0.203·97-s − 0.597·101-s + 0.197·103-s + 1.16·107-s + 0.191·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.050664257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.050664257\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.336176005892584946796187316207, −7.82938367544527042838083386422, −7.37070241903959494763217343062, −6.16479306435173437266649381708, −5.36122859561824758187462287059, −5.05140166391539482711859408027, −3.78170050965974938007153268583, −3.06042177598208541130831106448, −1.98044836881357242479095298109, −0.864246656632210342942849163068,
0.864246656632210342942849163068, 1.98044836881357242479095298109, 3.06042177598208541130831106448, 3.78170050965974938007153268583, 5.05140166391539482711859408027, 5.36122859561824758187462287059, 6.16479306435173437266649381708, 7.37070241903959494763217343062, 7.82938367544527042838083386422, 8.336176005892584946796187316207
