| L(s) = 1 | − 3-s − 2·9-s + 11-s − 13-s − 6·17-s − 2·19-s + 6·23-s − 5·25-s + 5·27-s + 9·29-s + 4·31-s − 33-s + 2·37-s + 39-s − 6·41-s + 4·43-s + 6·47-s + 6·51-s + 2·57-s + 3·59-s + 11·61-s − 11·67-s − 6·69-s + 2·73-s + 5·75-s − 5·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.301·11-s − 0.277·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s − 25-s + 0.962·27-s + 1.67·29-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.875·47-s + 0.840·51-s + 0.264·57-s + 0.390·59-s + 1.40·61-s − 1.34·67-s − 0.722·69-s + 0.234·73-s + 0.577·75-s − 0.562·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
| 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
−7.23125371039698292443632360161, −6.65241472062653308222612416481, −6.14853718702059937895066567222, −5.34931737834486556960138497960, −4.66196038971614403585102351941, −4.05923614656913661394578603646, −2.91027296261784789907385109519, −2.33677433521244575835493915149, −1.07055595102177968762291052173, 0,
1.07055595102177968762291052173, 2.33677433521244575835493915149, 2.91027296261784789907385109519, 4.05923614656913661394578603646, 4.66196038971614403585102351941, 5.34931737834486556960138497960, 6.14853718702059937895066567222, 6.65241472062653308222612416481, 7.23125371039698292443632360161
