| L(s) = 1 | − 3·3-s − 5-s − 7-s + 6·9-s − 4·11-s − 13-s + 3·15-s − 7·17-s + 19-s + 3·21-s − 5·23-s + 25-s − 9·27-s − 7·29-s − 2·31-s + 12·33-s + 35-s + 6·37-s + 3·39-s + 6·41-s − 10·43-s − 6·45-s − 8·47-s − 6·49-s + 21·51-s + 3·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s − 1.20·11-s − 0.277·13-s + 0.774·15-s − 1.69·17-s + 0.229·19-s + 0.654·21-s − 1.04·23-s + 1/5·25-s − 1.73·27-s − 1.29·29-s − 0.359·31-s + 2.08·33-s + 0.169·35-s + 0.986·37-s + 0.480·39-s + 0.937·41-s − 1.52·43-s − 0.894·45-s − 1.16·47-s − 6/7·49-s + 2.94·51-s + 0.412·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.27607453140199563799200853130, −6.49647606098543604685589245023, −5.96489830914417748928699663757, −5.23021974470597773591242547129, −4.61301216490531745542615203516, −3.94673179929506657329609690132, −2.73221613774994623011314064378, −1.64923211694522005871920996448, 0, 0,
1.64923211694522005871920996448, 2.73221613774994623011314064378, 3.94673179929506657329609690132, 4.61301216490531745542615203516, 5.23021974470597773591242547129, 5.96489830914417748928699663757, 6.49647606098543604685589245023, 7.27607453140199563799200853130
