| L(s) = 1 | − 2·3-s − 2·5-s + 2·9-s + 4·11-s − 2·13-s + 4·15-s − 2·17-s + 4·19-s + 2·23-s − 2·25-s − 6·27-s + 8·29-s − 12·31-s − 8·33-s + 2·37-s + 4·39-s + 6·41-s + 12·43-s − 4·45-s − 8·47-s − 14·49-s + 4·51-s − 4·53-s − 8·55-s − 8·57-s − 12·59-s − 20·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 2/3·9-s + 1.20·11-s − 0.554·13-s + 1.03·15-s − 0.485·17-s + 0.917·19-s + 0.417·23-s − 2/5·25-s − 1.15·27-s + 1.48·29-s − 2.15·31-s − 1.39·33-s + 0.328·37-s + 0.640·39-s + 0.937·41-s + 1.82·43-s − 0.596·45-s − 1.16·47-s − 2·49-s + 0.560·51-s − 0.549·53-s − 1.07·55-s − 1.05·57-s − 1.56·59-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12503296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12503296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 202 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.070423751989355262894536447711, −8.012849638758794544502949300618, −7.63343147220259371650111419074, −7.19268606758021770673816813323, −6.99202618464141443531216578306, −6.43739032450406368104465105039, −6.01741207628541299537386815801, −5.99627944341366461294265652969, −5.30961689986339759181572694857, −4.95234504415681896525416753919, −4.53836017009520124087892123851, −4.22818223806052836178265326085, −3.82407483186398511195019856375, −3.27864562959716462363730254626, −2.94339265111089655868009019039, −2.20480040132337849638982071540, −1.41770177720073088413420148690, −1.23515289458907998137356667077, 0, 0,
1.23515289458907998137356667077, 1.41770177720073088413420148690, 2.20480040132337849638982071540, 2.94339265111089655868009019039, 3.27864562959716462363730254626, 3.82407483186398511195019856375, 4.22818223806052836178265326085, 4.53836017009520124087892123851, 4.95234504415681896525416753919, 5.30961689986339759181572694857, 5.99627944341366461294265652969, 6.01741207628541299537386815801, 6.43739032450406368104465105039, 6.99202618464141443531216578306, 7.19268606758021770673816813323, 7.63343147220259371650111419074, 8.012849638758794544502949300618, 8.070423751989355262894536447711
