| L(s) = 1 | − 3-s − 2·7-s − 2·9-s − 11-s − 4·13-s + 2·17-s + 2·21-s − 23-s + 5·27-s − 7·31-s + 33-s − 3·37-s + 4·39-s − 8·41-s − 6·43-s + 8·47-s − 3·49-s − 2·51-s + 6·53-s − 5·59-s + 12·61-s + 4·63-s − 7·67-s + 69-s + 3·71-s − 4·73-s + 2·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s − 0.301·11-s − 1.10·13-s + 0.485·17-s + 0.436·21-s − 0.208·23-s + 0.962·27-s − 1.25·31-s + 0.174·33-s − 0.493·37-s + 0.640·39-s − 1.24·41-s − 0.914·43-s + 1.16·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s − 0.650·59-s + 1.53·61-s + 0.503·63-s − 0.855·67-s + 0.120·69-s + 0.356·71-s − 0.468·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6524026244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6524026244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| 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
−8.394238320905654147490500873996, −7.49534540311409429819739808064, −6.89747784534263149853829967422, −6.10845204313762133714030223921, −5.40055166000655653433368694834, −4.88140290332121063087048287423, −3.68709007429289538959412107978, −2.96905370396904801865789065276, −2.00337434233232539103171654262, −0.44102589544325217691153494662,
0.44102589544325217691153494662, 2.00337434233232539103171654262, 2.96905370396904801865789065276, 3.68709007429289538959412107978, 4.88140290332121063087048287423, 5.40055166000655653433368694834, 6.10845204313762133714030223921, 6.89747784534263149853829967422, 7.49534540311409429819739808064, 8.394238320905654147490500873996
