L(s) = 1 | − 4-s − 9-s − 4·11-s + 16-s − 8·19-s + 20·31-s + 36-s + 22·41-s + 4·44-s − 2·49-s + 30·59-s − 2·61-s − 64-s + 22·71-s + 8·76-s + 16·79-s + 81-s + 12·89-s + 4·99-s + 36·101-s + 28·109-s − 10·121-s − 20·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1/3·9-s − 1.20·11-s + 1/4·16-s − 1.83·19-s + 3.59·31-s + 1/6·36-s + 3.43·41-s + 0.603·44-s − 2/7·49-s + 3.90·59-s − 0.256·61-s − 1/8·64-s + 2.61·71-s + 0.917·76-s + 1.80·79-s + 1/9·81-s + 1.27·89-s + 0.402·99-s + 3.58·101-s + 2.68·109-s − 0.909·121-s − 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.132135841\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.132135841\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 141 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.979641706963142822271586399806, −8.546008060917776011503918865748, −8.292336160259142959544631054070, −8.208234869543162599507136360856, −7.48282601625325442294587195209, −7.45324991615617745713253489975, −6.64426613762148612579762386462, −6.22400587899111351619107430839, −6.18577458303303873598219905120, −5.63503650627466803185498972449, −4.98796752048631405859558859072, −4.75041175774852964809176751631, −4.50602854023142110750108078345, −3.72711382141726017625883179466, −3.65534335159504863052349560232, −2.64596247974211042143188680008, −2.40385887975412528415758913542, −2.22956103492190910402914332489, −0.76913164906147697318826679636, −0.74134550664436845406359186341,
0.74134550664436845406359186341, 0.76913164906147697318826679636, 2.22956103492190910402914332489, 2.40385887975412528415758913542, 2.64596247974211042143188680008, 3.65534335159504863052349560232, 3.72711382141726017625883179466, 4.50602854023142110750108078345, 4.75041175774852964809176751631, 4.98796752048631405859558859072, 5.63503650627466803185498972449, 6.18577458303303873598219905120, 6.22400587899111351619107430839, 6.64426613762148612579762386462, 7.45324991615617745713253489975, 7.48282601625325442294587195209, 8.208234869543162599507136360856, 8.292336160259142959544631054070, 8.546008060917776011503918865748, 8.979641706963142822271586399806