| L(s) = 1 | − 2·3-s + 9-s + 4·11-s + 6·13-s + 4·17-s + 6·19-s − 4·23-s + 4·27-s + 6·29-s + 4·31-s − 8·33-s − 6·37-s − 12·39-s + 4·41-s − 12·43-s − 12·47-s − 8·51-s + 6·53-s − 12·57-s + 6·59-s + 6·61-s − 12·67-s + 8·69-s − 8·71-s − 11·81-s − 6·83-s − 12·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.970·17-s + 1.37·19-s − 0.834·23-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 1.39·33-s − 0.986·37-s − 1.92·39-s + 0.624·41-s − 1.82·43-s − 1.75·47-s − 1.12·51-s + 0.824·53-s − 1.58·57-s + 0.781·59-s + 0.768·61-s − 1.46·67-s + 0.963·69-s − 0.949·71-s − 1.22·81-s − 0.658·83-s − 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.071259689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.071259689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.95506721540103, −13.60394789052215, −12.99911666418020, −12.27982125825683, −11.77282426153017, −11.70431513222914, −11.24994886789568, −10.48964406675283, −10.06888584744003, −9.718529176488667, −8.836377483030657, −8.485320504095633, −8.003205511359896, −7.128771671301716, −6.692957287596723, −6.173566024349264, −5.813794295442145, −5.249068381453410, −4.664577604041603, −3.946002054882837, −3.407174227929135, −2.910790862584769, −1.567541411706209, −1.276765589963390, −0.5748897167084592,
0.5748897167084592, 1.276765589963390, 1.567541411706209, 2.910790862584769, 3.407174227929135, 3.946002054882837, 4.664577604041603, 5.249068381453410, 5.813794295442145, 6.173566024349264, 6.692957287596723, 7.128771671301716, 8.003205511359896, 8.485320504095633, 8.836377483030657, 9.718529176488667, 10.06888584744003, 10.48964406675283, 11.24994886789568, 11.70431513222914, 11.77282426153017, 12.27982125825683, 12.99911666418020, 13.60394789052215, 13.95506721540103
