| L(s) = 1 | − 12·13-s + 8·25-s + 32·31-s − 28·37-s + 16·43-s + 48·61-s − 32·67-s − 12·73-s + 20·97-s − 16·103-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 3.32·13-s + 8/5·25-s + 5.74·31-s − 4.60·37-s + 2.43·43-s + 6.14·61-s − 3.90·67-s − 1.40·73-s + 2.03·97-s − 1.57·103-s − 1.81·121-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 + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242448438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242448438\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 4174 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.47610788582990476562674975480, −5.94021537829048781638457408456, −5.77208687405961952135496594669, −5.51736631244696594508427360856, −5.21828964152662605808944617819, −5.17101338678552941338752384318, −5.06021930871127566272128805278, −4.87351948303845375843259535202, −4.51769034409821509587442521322, −4.46419195160507331728356325473, −4.14480398200836247782085910109, −4.05064941840210235489053334655, −3.80056380607333870835486631893, −3.34997067269643435022578263970, −2.97290567139225049658401127234, −2.88969042146603586400369992828, −2.72159775541896938088305453121, −2.70966931649817523656649847434, −2.21314163964743257704286779755, −2.08632734781656558586723080781, −1.81004628486891877929003093542, −1.17106132764155026315180304810, −1.04756036353435138279563971982, −0.72061640829780883741166664823, −0.19660500810475677996352332051,
0.19660500810475677996352332051, 0.72061640829780883741166664823, 1.04756036353435138279563971982, 1.17106132764155026315180304810, 1.81004628486891877929003093542, 2.08632734781656558586723080781, 2.21314163964743257704286779755, 2.70966931649817523656649847434, 2.72159775541896938088305453121, 2.88969042146603586400369992828, 2.97290567139225049658401127234, 3.34997067269643435022578263970, 3.80056380607333870835486631893, 4.05064941840210235489053334655, 4.14480398200836247782085910109, 4.46419195160507331728356325473, 4.51769034409821509587442521322, 4.87351948303845375843259535202, 5.06021930871127566272128805278, 5.17101338678552941338752384318, 5.21828964152662605808944617819, 5.51736631244696594508427360856, 5.77208687405961952135496594669, 5.94021537829048781638457408456, 6.47610788582990476562674975480
