| L(s) = 1 | − 2·5-s − 5·7-s + 6·11-s − 13-s + 3·17-s − 5·19-s − 2·23-s − 7·25-s + 9·29-s − 4·31-s + 10·35-s − 5·37-s + 7·41-s − 3·43-s + 8·47-s + 18·49-s + 9·53-s − 12·55-s − 4·59-s − 2·61-s + 2·65-s − 12·67-s − 16·71-s + 13·73-s − 30·77-s − 8·79-s − 13·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.88·7-s + 1.80·11-s − 0.277·13-s + 0.727·17-s − 1.14·19-s − 0.417·23-s − 7/5·25-s + 1.67·29-s − 0.718·31-s + 1.69·35-s − 0.821·37-s + 1.09·41-s − 0.457·43-s + 1.16·47-s + 18/7·49-s + 1.23·53-s − 1.61·55-s − 0.520·59-s − 0.256·61-s + 0.248·65-s − 1.46·67-s − 1.89·71-s + 1.52·73-s − 3.41·77-s − 0.900·79-s − 1.42·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7872603130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7872603130\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 3 T - 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 13 T + 86 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.881270935275064878716957596930, −9.079544402297958354006976135980, −8.857707793223250808755212040489, −8.782469863475286379652677179194, −8.063683132379836087955773374077, −7.47392331068586121008046230826, −7.31566825169507251832636232699, −6.88858380727272206122658968619, −6.29866622011716750690264289548, −6.09797342313201400398079315702, −5.88928360775376908759111551739, −5.10318118359659614491679834001, −4.33940064343357763927277691063, −4.03207950475789621943506294131, −3.83820997555562842987112505327, −3.27579745282555623266205744600, −2.79283942078935197655726175668, −2.10393135698600217566451032853, −1.27103909602568448564728803852, −0.37518655202211718665225645374,
0.37518655202211718665225645374, 1.27103909602568448564728803852, 2.10393135698600217566451032853, 2.79283942078935197655726175668, 3.27579745282555623266205744600, 3.83820997555562842987112505327, 4.03207950475789621943506294131, 4.33940064343357763927277691063, 5.10318118359659614491679834001, 5.88928360775376908759111551739, 6.09797342313201400398079315702, 6.29866622011716750690264289548, 6.88858380727272206122658968619, 7.31566825169507251832636232699, 7.47392331068586121008046230826, 8.063683132379836087955773374077, 8.782469863475286379652677179194, 8.857707793223250808755212040489, 9.079544402297958354006976135980, 9.881270935275064878716957596930
