| L(s) = 1 | − 4-s − 8·11-s + 16-s − 12·19-s + 12·29-s − 8·31-s + 20·41-s + 8·44-s − 49-s + 8·59-s − 16·61-s − 64-s − 4·71-s + 12·76-s − 32·79-s − 4·89-s − 12·101-s + 36·109-s − 12·116-s + 26·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.41·11-s + 1/4·16-s − 2.75·19-s + 2.22·29-s − 1.43·31-s + 3.12·41-s + 1.20·44-s − 1/7·49-s + 1.04·59-s − 2.04·61-s − 1/8·64-s − 0.474·71-s + 1.37·76-s − 3.60·79-s − 0.423·89-s − 1.19·101-s + 3.44·109-s − 1.11·116-s + 2.36·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2548044428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2548044428\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.728574922293480237348553011838, −8.663187105128396370217008490890, −8.160618847450276972319110530546, −7.61865278886682822730325139366, −7.61621120401563747472151358691, −7.14694670240559627966143056293, −6.43161845270593378323672486691, −6.28220007064314980836071518760, −5.67488746212090373125868442909, −5.61545149263463596047728862679, −4.86627943848617486282116948084, −4.68397690188515935167409193995, −4.16594551363586349195500193966, −4.05239657424506183186052804338, −3.06729337701200944141762454417, −2.81251756406472609130400871824, −2.39856718184895471042088853610, −1.95996452690452873532812760320, −1.07983564110204276269333971784, −0.16961349179264375423260645715,
0.16961349179264375423260645715, 1.07983564110204276269333971784, 1.95996452690452873532812760320, 2.39856718184895471042088853610, 2.81251756406472609130400871824, 3.06729337701200944141762454417, 4.05239657424506183186052804338, 4.16594551363586349195500193966, 4.68397690188515935167409193995, 4.86627943848617486282116948084, 5.61545149263463596047728862679, 5.67488746212090373125868442909, 6.28220007064314980836071518760, 6.43161845270593378323672486691, 7.14694670240559627966143056293, 7.61621120401563747472151358691, 7.61865278886682822730325139366, 8.160618847450276972319110530546, 8.663187105128396370217008490890, 8.728574922293480237348553011838
