| L(s) = 1 | + 9·13-s − 8·19-s + 5·25-s + 14·31-s + 3·43-s + 11·49-s − 61-s + 11·67-s + 17·73-s − 17·79-s + 24·97-s − 14·103-s + 36·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 41·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.49·13-s − 1.83·19-s + 25-s + 2.51·31-s + 0.457·43-s + 11/7·49-s − 0.128·61-s + 1.34·67-s + 1.98·73-s − 1.91·79-s + 2.43·97-s − 1.37·103-s + 3.44·109-s + 2·121-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 + 3.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.610808825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.610808825\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−8.719140888055867441608011788356, −8.551437021430137707522251499780, −8.306474864771470966562884629388, −8.295281912548680510010605338664, −7.35411392181660839900008508374, −7.18380769215877349521827336958, −6.48565859301277859419015297691, −6.43428391563242308233867846327, −5.93118469062646253544049786026, −5.83454040499677930876511298497, −5.03376896420342561840532107718, −4.65295640390820777370049660403, −4.12763693954251826797621201517, −4.03817414099960516379896148426, −3.19214276063401651569658488838, −3.13815437477491998902486821030, −2.20334430234549329988242126216, −1.98287532879091870427986315871, −0.910470052617604753593672075841, −0.855825255866622988987925164973,
0.855825255866622988987925164973, 0.910470052617604753593672075841, 1.98287532879091870427986315871, 2.20334430234549329988242126216, 3.13815437477491998902486821030, 3.19214276063401651569658488838, 4.03817414099960516379896148426, 4.12763693954251826797621201517, 4.65295640390820777370049660403, 5.03376896420342561840532107718, 5.83454040499677930876511298497, 5.93118469062646253544049786026, 6.43428391563242308233867846327, 6.48565859301277859419015297691, 7.18380769215877349521827336958, 7.35411392181660839900008508374, 8.295281912548680510010605338664, 8.306474864771470966562884629388, 8.551437021430137707522251499780, 8.719140888055867441608011788356
