| L(s) = 1 | + 2-s − 7-s − 8-s + 3·11-s + 5·13-s − 14-s − 16-s − 3·17-s + 19-s + 3·22-s − 6·23-s − 10·25-s + 5·26-s + 3·29-s + 4·31-s − 3·34-s + 10·37-s + 38-s + 3·41-s + 4·43-s − 6·46-s − 18·47-s − 10·50-s − 18·53-s + 56-s + 3·58-s − 12·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.377·7-s − 0.353·8-s + 0.904·11-s + 1.38·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s + 0.229·19-s + 0.639·22-s − 1.25·23-s − 2·25-s + 0.980·26-s + 0.557·29-s + 0.718·31-s − 0.514·34-s + 1.64·37-s + 0.162·38-s + 0.468·41-s + 0.609·43-s − 0.884·46-s − 2.62·47-s − 1.41·50-s − 2.47·53-s + 0.133·56-s + 0.393·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.424129732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.424129732\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 4 T - 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.610462868568052418716175742451, −9.388937179498135507382387684665, −8.737990076817477109617846488716, −8.226914357979426649871526786383, −8.183324180401410133677717161392, −7.61683149940659480528452757406, −7.14233324796950389676105720574, −6.40836203800646656378969807257, −6.30249244750848115376315042536, −5.89141594044737731567491294008, −5.87164529579673870934697256038, −4.77412427150381684083377398104, −4.58397341465842262585642988379, −4.14698513640900012549366705193, −3.75618257643458404464783175866, −3.17185886931246411282958164075, −2.89928165904639548972601996525, −1.83455413428564013820988224554, −1.64573599960851079880260381742, −0.53053753308917060186029547312,
0.53053753308917060186029547312, 1.64573599960851079880260381742, 1.83455413428564013820988224554, 2.89928165904639548972601996525, 3.17185886931246411282958164075, 3.75618257643458404464783175866, 4.14698513640900012549366705193, 4.58397341465842262585642988379, 4.77412427150381684083377398104, 5.87164529579673870934697256038, 5.89141594044737731567491294008, 6.30249244750848115376315042536, 6.40836203800646656378969807257, 7.14233324796950389676105720574, 7.61683149940659480528452757406, 8.183324180401410133677717161392, 8.226914357979426649871526786383, 8.737990076817477109617846488716, 9.388937179498135507382387684665, 9.610462868568052418716175742451
