| L(s) = 1 | − 3·9-s + 14·13-s + 10·25-s − 2·37-s + 28·61-s + 14·73-s + 9·81-s − 28·97-s − 34·109-s − 42·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 121·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 9-s + 3.88·13-s + 2·25-s − 0.328·37-s + 3.58·61-s + 1.63·73-s + 81-s − 2.84·97-s − 3.25·109-s − 3.88·117-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 + 9.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-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
| \(L(1)\) |
\(\approx\) |
\(3.323838826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.323838826\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.235978052144681948681978355269, −8.662860358364950484133407084626, −8.384998978592289281633901685596, −8.170107535664199386557450119400, −8.014814095905459678834855941958, −6.97566821064394286895211477384, −6.72052141576389241610331893337, −6.62886201628499999644719600026, −5.96049951223975620415082524032, −5.76719395592021258873440951344, −5.27960588827974201800863393800, −4.99599427830733912744583560435, −4.12467184281138838191689878433, −3.71259627038352938953321614655, −3.70721667549705952982994994509, −2.90982759903641038627608475283, −2.64893838576950113212914994972, −1.72507014875236128408896494049, −1.16568221806449474142196239590, −0.73733607608865045705519489337,
0.73733607608865045705519489337, 1.16568221806449474142196239590, 1.72507014875236128408896494049, 2.64893838576950113212914994972, 2.90982759903641038627608475283, 3.70721667549705952982994994509, 3.71259627038352938953321614655, 4.12467184281138838191689878433, 4.99599427830733912744583560435, 5.27960588827974201800863393800, 5.76719395592021258873440951344, 5.96049951223975620415082524032, 6.62886201628499999644719600026, 6.72052141576389241610331893337, 6.97566821064394286895211477384, 8.014814095905459678834855941958, 8.170107535664199386557450119400, 8.384998978592289281633901685596, 8.662860358364950484133407084626, 9.235978052144681948681978355269
