| L(s) = 1 | + 3-s + 4-s − 2·7-s − 2·9-s + 12-s + 10·13-s + 16-s + 2·19-s − 2·21-s − 10·25-s − 5·27-s − 2·28-s − 8·31-s − 2·36-s + 4·37-s + 10·39-s + 16·43-s + 48-s − 11·49-s + 10·52-s + 2·57-s − 20·61-s + 4·63-s + 64-s + 10·67-s − 14·73-s − 10·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.755·7-s − 2/3·9-s + 0.288·12-s + 2.77·13-s + 1/4·16-s + 0.458·19-s − 0.436·21-s − 2·25-s − 0.962·27-s − 0.377·28-s − 1.43·31-s − 1/3·36-s + 0.657·37-s + 1.60·39-s + 2.43·43-s + 0.144·48-s − 1.57·49-s + 1.38·52-s + 0.264·57-s − 2.56·61-s + 0.503·63-s + 1/8·64-s + 1.22·67-s − 1.63·73-s − 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.312736779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.312736779\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−11.24807629195878921565630077767, −10.98048210666359838390472921844, −10.09897479975324709020702728875, −9.572003756910085560364635840628, −8.872794307339397941412432360955, −8.684413720393242419277779815238, −7.73591843742988285704805588989, −7.53492665725128185091116505938, −6.40690082764165790518888770884, −5.85342474444132611353121181432, −5.81027585652647529822118508915, −4.18290710455057630899406360415, −3.56295849457415925614690839234, −2.99432139105097413830284496966, −1.70600515778630825145695828253,
1.70600515778630825145695828253, 2.99432139105097413830284496966, 3.56295849457415925614690839234, 4.18290710455057630899406360415, 5.81027585652647529822118508915, 5.85342474444132611353121181432, 6.40690082764165790518888770884, 7.53492665725128185091116505938, 7.73591843742988285704805588989, 8.684413720393242419277779815238, 8.872794307339397941412432360955, 9.572003756910085560364635840628, 10.09897479975324709020702728875, 10.98048210666359838390472921844, 11.24807629195878921565630077767
