| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 8-s + 9-s − 3·10-s − 11-s + 12-s + 6·13-s + 3·15-s + 16-s − 5·17-s − 18-s + 6·19-s + 3·20-s + 22-s + 5·23-s − 24-s + 4·25-s − 6·26-s + 27-s − 6·29-s − 3·30-s + 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.301·11-s + 0.288·12-s + 1.66·13-s + 0.774·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 1.37·19-s + 0.670·20-s + 0.213·22-s + 1.04·23-s − 0.204·24-s + 4/5·25-s − 1.17·26-s + 0.192·27-s − 1.11·29-s − 0.547·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.419660610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.419660610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
| 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
−8.842237582261943412497659784450, −8.102581647527097612509052589247, −7.20820344916316378150903617194, −6.46756871669938820534669985051, −5.81123907221281022677407571078, −4.97628229745406794569182790979, −3.69490446332830124495370704994, −2.82218766450276993295029177486, −1.91855494693826267909351961029, −1.09688652548402126508340746432,
1.09688652548402126508340746432, 1.91855494693826267909351961029, 2.82218766450276993295029177486, 3.69490446332830124495370704994, 4.97628229745406794569182790979, 5.81123907221281022677407571078, 6.46756871669938820534669985051, 7.20820344916316378150903617194, 8.102581647527097612509052589247, 8.842237582261943412497659784450
