| L(s) = 1 | − 3·9-s + 11-s − 4·16-s + 8·23-s − 25-s − 7·49-s + 20·53-s + 28·67-s + 16·71-s − 3·99-s + 16·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 17·169-s + 173-s − 4·176-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 9-s + 0.301·11-s − 16-s + 1.66·23-s − 1/5·25-s − 49-s + 2.74·53-s + 3.42·67-s + 1.89·71-s − 0.301·99-s + 1.50·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.30·169-s + 0.0760·173-s − 0.301·176-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387801979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387801979\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
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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
−9.224414175459041687773963031127, −8.749546641820203741214562547446, −8.493580001103400656477947030685, −7.931979702871659852073370326371, −7.24006600412275574857535482217, −6.79445842090327170160582443731, −6.48006898691393493047746605363, −5.70193838984999107772234534861, −5.23975374025055338202416439938, −4.81112539523175410463567763555, −3.99095407742630478815800076171, −3.46110135681904694119673074968, −2.67138926466593006156481479333, −2.12629538117729852673534058825, −0.800312735555445643372264521122,
0.800312735555445643372264521122, 2.12629538117729852673534058825, 2.67138926466593006156481479333, 3.46110135681904694119673074968, 3.99095407742630478815800076171, 4.81112539523175410463567763555, 5.23975374025055338202416439938, 5.70193838984999107772234534861, 6.48006898691393493047746605363, 6.79445842090327170160582443731, 7.24006600412275574857535482217, 7.931979702871659852073370326371, 8.493580001103400656477947030685, 8.749546641820203741214562547446, 9.224414175459041687773963031127
