| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 2·5-s + 8·6-s − 2·7-s + 8·9-s + 4·10-s − 8·12-s − 8·13-s + 4·14-s + 8·15-s − 4·16-s + 4·17-s − 16·18-s + 4·19-s − 4·20-s + 8·21-s − 8·23-s − 25-s + 16·26-s − 12·27-s − 4·28-s − 16·30-s + 8·32-s − 8·34-s + 4·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s − 0.894·5-s + 3.26·6-s − 0.755·7-s + 8/3·9-s + 1.26·10-s − 2.30·12-s − 2.21·13-s + 1.06·14-s + 2.06·15-s − 16-s + 0.970·17-s − 3.77·18-s + 0.917·19-s − 0.894·20-s + 1.74·21-s − 1.66·23-s − 1/5·25-s + 3.13·26-s − 2.30·27-s − 0.755·28-s − 2.92·30-s + 1.41·32-s − 1.37·34-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−12.08468361701946296351947901692, −11.74232555403500297799651086096, −10.82321017671691156828651678220, −10.71756217274233987093872303941, −10.06476964818689916668247879107, −9.656433953158384843371413337775, −9.597462312647546564069275835363, −8.526512738328720310933327079795, −7.85972654731057360047242333339, −7.54182331162429303070473363800, −6.95985893794383838459495489488, −6.67731876598189408121292457164, −5.73192741945282116023582038379, −5.44283590408288886221109332831, −4.77350202420320666474275227815, −4.10408770385390336147271368246, −3.05343803685162763025309296745, −1.65791358384701540528400149726, 0, 0,
1.65791358384701540528400149726, 3.05343803685162763025309296745, 4.10408770385390336147271368246, 4.77350202420320666474275227815, 5.44283590408288886221109332831, 5.73192741945282116023582038379, 6.67731876598189408121292457164, 6.95985893794383838459495489488, 7.54182331162429303070473363800, 7.85972654731057360047242333339, 8.526512738328720310933327079795, 9.597462312647546564069275835363, 9.656433953158384843371413337775, 10.06476964818689916668247879107, 10.71756217274233987093872303941, 10.82321017671691156828651678220, 11.74232555403500297799651086096, 12.08468361701946296351947901692
