| L(s) = 1 | + 5-s + 4·7-s + 6·11-s − 4·13-s + 3·17-s + 7·19-s − 9·23-s + 25-s + 7·31-s + 4·35-s + 2·37-s − 6·41-s − 2·43-s + 9·49-s − 9·53-s + 6·55-s − 12·59-s − 7·61-s − 4·65-s − 2·67-s + 6·71-s + 2·73-s + 24·77-s + 79-s + 9·83-s + 3·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.80·11-s − 1.10·13-s + 0.727·17-s + 1.60·19-s − 1.87·23-s + 1/5·25-s + 1.25·31-s + 0.676·35-s + 0.328·37-s − 0.937·41-s − 0.304·43-s + 9/7·49-s − 1.23·53-s + 0.809·55-s − 1.56·59-s − 0.896·61-s − 0.496·65-s − 0.244·67-s + 0.712·71-s + 0.234·73-s + 2.73·77-s + 0.112·79-s + 0.987·83-s + 0.325·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.602632500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.602632500\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−9.258800425324802754863509016129, −8.092167622332816991340248763807, −7.74019567158184250231213816224, −6.71899526548996053710103103705, −5.89394292664879166824137012839, −4.99976597498701492766603898596, −4.36525567642916231248237875179, −3.26781961523861610322938673492, −1.93615831004412516494141020149, −1.21444767823394362882446466641,
1.21444767823394362882446466641, 1.93615831004412516494141020149, 3.26781961523861610322938673492, 4.36525567642916231248237875179, 4.99976597498701492766603898596, 5.89394292664879166824137012839, 6.71899526548996053710103103705, 7.74019567158184250231213816224, 8.092167622332816991340248763807, 9.258800425324802754863509016129
