| L(s) = 1 | − 7-s + 7·13-s − 8·19-s + 5·25-s + 22·31-s + 37-s + 13·43-s − 6·49-s − 2·61-s + 22·67-s + 10·73-s − 26·79-s − 7·91-s + 19·97-s + 7·103-s + 19·109-s + 11·121-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.94·13-s − 1.83·19-s + 25-s + 3.95·31-s + 0.164·37-s + 1.98·43-s − 6/7·49-s − 0.256·61-s + 2.68·67-s + 1.17·73-s − 2.92·79-s − 0.733·91-s + 1.92·97-s + 0.689·103-s + 1.81·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083406252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083406252\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−8.946105476706226060414424613546, −8.925338174027574405539712968071, −8.339337352969809099960705614591, −8.145568567701626386864554725086, −7.993711805353816369273444181396, −7.08458385393307574404798323208, −6.84074956637456655038376477234, −6.40932566044695748795286546898, −6.08675432134947216703489736872, −5.99259733611794745766892944916, −5.27393562435221367877755012961, −4.65437980615657097817887047300, −4.24443792208951395271962241392, −4.21259013862628706212730304448, −3.22078743837634814155001143002, −3.19788529700938336785067869900, −2.41704857296097390028232057219, −2.00369312998558517200722227640, −1.00800530234850543059297618034, −0.77401877661380594419359150400,
0.77401877661380594419359150400, 1.00800530234850543059297618034, 2.00369312998558517200722227640, 2.41704857296097390028232057219, 3.19788529700938336785067869900, 3.22078743837634814155001143002, 4.21259013862628706212730304448, 4.24443792208951395271962241392, 4.65437980615657097817887047300, 5.27393562435221367877755012961, 5.99259733611794745766892944916, 6.08675432134947216703489736872, 6.40932566044695748795286546898, 6.84074956637456655038376477234, 7.08458385393307574404798323208, 7.993711805353816369273444181396, 8.145568567701626386864554725086, 8.339337352969809099960705614591, 8.925338174027574405539712968071, 8.946105476706226060414424613546
