L(s) = 1 | − 2-s + 4-s + 6·5-s − 8-s − 6·10-s + 16-s + 4·19-s + 6·20-s − 12·23-s + 17·25-s + 12·29-s − 32-s − 4·38-s − 6·40-s − 20·43-s + 12·46-s + 12·47-s − 13·49-s − 17·50-s + 18·53-s − 12·58-s + 64-s + 28·67-s − 14·73-s + 4·76-s + 6·80-s + 20·86-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 2.68·5-s − 0.353·8-s − 1.89·10-s + 1/4·16-s + 0.917·19-s + 1.34·20-s − 2.50·23-s + 17/5·25-s + 2.22·29-s − 0.176·32-s − 0.648·38-s − 0.948·40-s − 3.04·43-s + 1.76·46-s + 1.75·47-s − 1.85·49-s − 2.40·50-s + 2.47·53-s − 1.57·58-s + 1/8·64-s + 3.42·67-s − 1.63·73-s + 0.458·76-s + 0.670·80-s + 2.15·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.808484862\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.808484862\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.872045258839871807563899217385, −9.300538826904107531692188360431, −8.699511810288277483701861071116, −8.268048909391662018895990050852, −7.82087500680797002813709866504, −6.73860900674537002760989556093, −6.68641118928175950907657181471, −6.10265864740654543835243104495, −5.43217621063437274714591238543, −5.37363015537505503550836847886, −4.34994968190190118318545746782, −3.35914372705473359723249826168, −2.44850978333263328730546301919, −2.05899421145709367031472603742, −1.26980488075185704480426069120,
1.26980488075185704480426069120, 2.05899421145709367031472603742, 2.44850978333263328730546301919, 3.35914372705473359723249826168, 4.34994968190190118318545746782, 5.37363015537505503550836847886, 5.43217621063437274714591238543, 6.10265864740654543835243104495, 6.68641118928175950907657181471, 6.73860900674537002760989556093, 7.82087500680797002813709866504, 8.268048909391662018895990050852, 8.699511810288277483701861071116, 9.300538826904107531692188360431, 9.872045258839871807563899217385