| L(s) = 1 | − 4-s + 16-s − 4·19-s − 8·31-s + 12·41-s + 10·49-s − 20·61-s − 64-s + 4·76-s − 16·79-s + 12·89-s + 24·101-s + 32·109-s − 22·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 12·164-s + 167-s − 169-s + 173-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s − 0.917·19-s − 1.43·31-s + 1.87·41-s + 10/7·49-s − 2.56·61-s − 1/8·64-s + 0.458·76-s − 1.80·79-s + 1.27·89-s + 2.38·101-s + 3.06·109-s − 2·121-s + 0.718·124-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 − 0.937·164-s + 0.0773·167-s − 0.0769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593115900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593115900\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.169385420287446907991853890329, −8.059826631698065938273243112727, −7.43689563544388306928317870698, −7.24219985745763752113086955983, −7.12212854074895949071130231258, −6.27983602961984556122515326997, −6.18481797254258035466347439644, −5.71960972394324912791380426165, −5.61231806324589921026656971107, −4.83002052795350724146455741050, −4.72236309063112340841944306961, −4.20372234779582980550909298256, −4.03074970426613406889338523094, −3.34602889051150886701336101414, −3.19151900049041085647312607639, −2.51705140556881454726675404655, −2.07908366386907650415466676821, −1.68215248756078875218895725144, −0.946340358144225596878288945148, −0.38349345436385057722912645966,
0.38349345436385057722912645966, 0.946340358144225596878288945148, 1.68215248756078875218895725144, 2.07908366386907650415466676821, 2.51705140556881454726675404655, 3.19151900049041085647312607639, 3.34602889051150886701336101414, 4.03074970426613406889338523094, 4.20372234779582980550909298256, 4.72236309063112340841944306961, 4.83002052795350724146455741050, 5.61231806324589921026656971107, 5.71960972394324912791380426165, 6.18481797254258035466347439644, 6.27983602961984556122515326997, 7.12212854074895949071130231258, 7.24219985745763752113086955983, 7.43689563544388306928317870698, 8.059826631698065938273243112727, 8.169385420287446907991853890329
