| L(s) = 1 | − 9-s + 6·11-s + 16·19-s + 18·29-s − 12·31-s − 16·41-s − 49-s + 24·61-s + 6·71-s + 10·79-s + 81-s + 8·89-s − 6·99-s + 20·101-s + 38·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 16·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.80·11-s + 3.67·19-s + 3.34·29-s − 2.15·31-s − 2.49·41-s − 1/7·49-s + 3.07·61-s + 0.712·71-s + 1.12·79-s + 1/9·81-s + 0.847·89-s − 0.603·99-s + 1.99·101-s + 3.63·109-s + 5/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 1.22·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.424567266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.424567266\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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
−8.704505674082592620260202309242, −8.187923369902152995166649863439, −7.898942400311169212589471007564, −7.43760147482563528734997427870, −6.99103526104187081149765880774, −6.91119132038125307939519233462, −6.35371314060254852111347145277, −6.20783711089548803512652476464, −5.37811539116601693211521907791, −5.33069016064040693192817048201, −4.96024628230441791229489511833, −4.56105153162822986179824101318, −3.77691290231479904664236781341, −3.53716107302570067133031187726, −3.35616352944409429704941473382, −2.83080029464948080733537017862, −2.17271349420402334196283852041, −1.58078702381415997420633069960, −1.00772827271505199697381032314, −0.76113048837905814381134759628,
0.76113048837905814381134759628, 1.00772827271505199697381032314, 1.58078702381415997420633069960, 2.17271349420402334196283852041, 2.83080029464948080733537017862, 3.35616352944409429704941473382, 3.53716107302570067133031187726, 3.77691290231479904664236781341, 4.56105153162822986179824101318, 4.96024628230441791229489511833, 5.33069016064040693192817048201, 5.37811539116601693211521907791, 6.20783711089548803512652476464, 6.35371314060254852111347145277, 6.91119132038125307939519233462, 6.99103526104187081149765880774, 7.43760147482563528734997427870, 7.898942400311169212589471007564, 8.187923369902152995166649863439, 8.704505674082592620260202309242
