| L(s) = 1 | − 2·7-s + 12·17-s + 16·23-s + 6·25-s − 16·31-s − 12·41-s + 24·47-s + 3·49-s + 12·73-s + 12·89-s − 20·97-s + 8·103-s + 36·113-s − 24·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 32·161-s + 163-s + 167-s − 10·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2.91·17-s + 3.33·23-s + 6/5·25-s − 2.87·31-s − 1.87·41-s + 3.50·47-s + 3/7·49-s + 1.40·73-s + 1.27·89-s − 2.03·97-s + 0.788·103-s + 3.38·113-s − 2.20·119-s + 1.63·121-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 − 2.52·161-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.605565861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.605565861\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.808403309351388672536331540702, −8.337934849524808270569581995651, −7.64681512079672486673555495127, −7.55160282233112784942132997330, −7.09630123037333113588879970789, −6.94320062609902561388226159792, −6.61921990279707123021578868475, −5.75024090520893590247927042600, −5.56526152050135321541992395755, −5.54885722870958506325415901972, −4.84237008492036532612821226815, −4.70340125006679818102810707053, −3.74506568742597022604786064860, −3.58840272941922389825174386599, −3.10194262993791376912793064246, −3.04970745004688168731530314786, −2.27268346840860367279736915015, −1.62193942419718725764356837302, −0.909230652591034306570431116367, −0.73827499140092141964850160956,
0.73827499140092141964850160956, 0.909230652591034306570431116367, 1.62193942419718725764356837302, 2.27268346840860367279736915015, 3.04970745004688168731530314786, 3.10194262993791376912793064246, 3.58840272941922389825174386599, 3.74506568742597022604786064860, 4.70340125006679818102810707053, 4.84237008492036532612821226815, 5.54885722870958506325415901972, 5.56526152050135321541992395755, 5.75024090520893590247927042600, 6.61921990279707123021578868475, 6.94320062609902561388226159792, 7.09630123037333113588879970789, 7.55160282233112784942132997330, 7.64681512079672486673555495127, 8.337934849524808270569581995651, 8.808403309351388672536331540702
