| L(s) = 1 | − 2·3-s + 3·9-s + 4·19-s + 7·25-s − 4·27-s − 18·29-s + 2·31-s − 4·37-s + 18·53-s − 8·57-s − 6·59-s − 14·75-s + 5·81-s − 30·83-s + 36·87-s − 4·93-s + 8·103-s − 8·109-s + 8·111-s − 12·113-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 0.917·19-s + 7/5·25-s − 0.769·27-s − 3.34·29-s + 0.359·31-s − 0.657·37-s + 2.47·53-s − 1.05·57-s − 0.781·59-s − 1.61·75-s + 5/9·81-s − 3.29·83-s + 3.85·87-s − 0.414·93-s + 0.788·103-s − 0.766·109-s + 0.759·111-s − 1.12·113-s − 0.454·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9598221381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9598221381\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 155 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
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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
−9.399313986225159804161803806916, −8.689566754130849929814897972891, −8.619054464138582699759206127593, −7.87714954488260912208687688887, −7.44570103384186901012538675461, −7.23480578719337984755746912612, −6.94111526070505957458140205928, −6.46142402800890392576405098808, −5.93059242114323604959642214856, −5.54110775059362398383594301348, −5.41254595532140251450104078766, −4.95379099453019211167493133182, −4.41899804655451168354070894288, −3.91557220328210265752888404067, −3.62439317450712667913210896472, −2.96422302701552219325839491158, −2.41829803001243573852728731649, −1.66366257881429197187179758114, −1.24236527673125871696258896508, −0.38878806783810151063413324181,
0.38878806783810151063413324181, 1.24236527673125871696258896508, 1.66366257881429197187179758114, 2.41829803001243573852728731649, 2.96422302701552219325839491158, 3.62439317450712667913210896472, 3.91557220328210265752888404067, 4.41899804655451168354070894288, 4.95379099453019211167493133182, 5.41254595532140251450104078766, 5.54110775059362398383594301348, 5.93059242114323604959642214856, 6.46142402800890392576405098808, 6.94111526070505957458140205928, 7.23480578719337984755746912612, 7.44570103384186901012538675461, 7.87714954488260912208687688887, 8.619054464138582699759206127593, 8.689566754130849929814897972891, 9.399313986225159804161803806916
