| L(s) = 1 | − 3·3-s + 2·5-s − 2·7-s + 2·9-s + 11-s − 2·13-s − 6·15-s + 17-s + 2·19-s + 6·21-s + 2·23-s − 7·25-s + 6·27-s + 5·29-s + 31-s − 3·33-s − 4·35-s − 14·37-s + 6·39-s + 9·41-s − 8·43-s + 4·45-s − 6·47-s + 3·49-s − 3·51-s + 3·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s − 0.755·7-s + 2/3·9-s + 0.301·11-s − 0.554·13-s − 1.54·15-s + 0.242·17-s + 0.458·19-s + 1.30·21-s + 0.417·23-s − 7/5·25-s + 1.15·27-s + 0.928·29-s + 0.179·31-s − 0.522·33-s − 0.676·35-s − 2.30·37-s + 0.960·39-s + 1.40·41-s − 1.21·43-s + 0.596·45-s − 0.875·47-s + 3/7·49-s − 0.420·51-s + 0.412·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4528384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4528384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - T - 39 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 103 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 97 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 20 T + 213 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 103 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 101 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 97 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 197 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 183 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.839575968020472540194575750239, −8.833983926835018067458552718470, −7.899184365174266849623141662943, −7.78700496104395038634388329439, −7.14164500255635955891571988888, −6.76956516264692419252188916430, −6.30870163837418532801635703302, −6.21507577375541435408955903766, −5.66256735557282101752380578083, −5.53323163616067750542999721510, −4.91711529117347146996794830065, −4.79041835012383709587946987796, −4.03470858277818001569700588155, −3.52551643625639084960069568012, −2.85931159964067051249689273960, −2.63799976163539761420749992768, −1.59673768418762382821840388724, −1.32812191346529978745892661601, 0, 0,
1.32812191346529978745892661601, 1.59673768418762382821840388724, 2.63799976163539761420749992768, 2.85931159964067051249689273960, 3.52551643625639084960069568012, 4.03470858277818001569700588155, 4.79041835012383709587946987796, 4.91711529117347146996794830065, 5.53323163616067750542999721510, 5.66256735557282101752380578083, 6.21507577375541435408955903766, 6.30870163837418532801635703302, 6.76956516264692419252188916430, 7.14164500255635955891571988888, 7.78700496104395038634388329439, 7.899184365174266849623141662943, 8.833983926835018067458552718470, 8.839575968020472540194575750239
