| L(s) = 1 | − 4·7-s + 8·19-s + 10·25-s − 12·29-s + 8·31-s + 4·37-s + 9·49-s − 12·53-s + 24·59-s + 24·83-s + 8·103-s + 20·109-s + 12·113-s + 10·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s − 40·175-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.83·19-s + 2·25-s − 2.22·29-s + 1.43·31-s + 0.657·37-s + 9/7·49-s − 1.64·53-s + 3.12·59-s + 2.63·83-s + 0.788·103-s + 1.91·109-s + 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-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 + 0.0760·173-s − 3.02·175-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
| \(L(1)\) |
\(\approx\) |
\(2.588544701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.588544701\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−8.729042830694378557166007395628, −8.248255019414782111912352080542, −7.73354841966454197465585841455, −7.60421089041191712934912239510, −7.09079297781628177176323454343, −6.79381908308950977354078845526, −6.43564900400331234984549772955, −6.18747274383979713621610425701, −5.52796586846360439417760801881, −5.41854346842618749203109540012, −4.92799333860470475965942300229, −4.50334866010050679100304505693, −3.92320028349439863829764838679, −3.49587098587077795521985227194, −3.20728904815708636043613575844, −2.86827251153802823377210104329, −2.33926009880233913779483369028, −1.72722559933542220662194048639, −0.850528696633354334791286061642, −0.62770340494346190445203198598,
0.62770340494346190445203198598, 0.850528696633354334791286061642, 1.72722559933542220662194048639, 2.33926009880233913779483369028, 2.86827251153802823377210104329, 3.20728904815708636043613575844, 3.49587098587077795521985227194, 3.92320028349439863829764838679, 4.50334866010050679100304505693, 4.92799333860470475965942300229, 5.41854346842618749203109540012, 5.52796586846360439417760801881, 6.18747274383979713621610425701, 6.43564900400331234984549772955, 6.79381908308950977354078845526, 7.09079297781628177176323454343, 7.60421089041191712934912239510, 7.73354841966454197465585841455, 8.248255019414782111912352080542, 8.729042830694378557166007395628
