| L(s) = 1 | − 3·2-s + 4·4-s − 3·8-s + 3·16-s − 12·17-s − 4·19-s − 6·23-s − 8·31-s − 6·32-s + 36·34-s + 12·38-s + 18·46-s − 14·49-s − 24·53-s − 2·61-s + 24·62-s + 5·64-s − 48·68-s − 16·76-s − 16·79-s − 6·83-s − 24·92-s + 42·98-s + 72·106-s + 14·109-s − 22·121-s + 6·122-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s − 1.06·8-s + 3/4·16-s − 2.91·17-s − 0.917·19-s − 1.25·23-s − 1.43·31-s − 1.06·32-s + 6.17·34-s + 1.94·38-s + 2.65·46-s − 2·49-s − 3.29·53-s − 0.256·61-s + 3.04·62-s + 5/8·64-s − 5.82·68-s − 1.83·76-s − 1.80·79-s − 0.658·83-s − 2.50·92-s + 4.24·98-s + 6.99·106-s + 1.34·109-s − 2·121-s + 0.543·122-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 24 T + 245 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−9.989660451384092991567192162963, −9.846806652314932081194942554342, −9.262817905128557648662036611928, −9.063032592114854685808495850440, −8.541549067508522697176521220742, −8.430775478361451180134409531171, −7.70561925256738231799922053064, −7.62146689440114469921833710882, −6.73340307970039576492244038628, −6.53745925572367705019854657846, −6.12043055137277751861825322723, −5.40736753766768044370573974354, −4.49125527143314244312390003472, −4.43930060989674309234358785788, −3.54067116349947058101732812041, −2.76704267627356110074568601757, −1.87757341903363932694788708409, −1.69513718397245009221846235026, 0, 0,
1.69513718397245009221846235026, 1.87757341903363932694788708409, 2.76704267627356110074568601757, 3.54067116349947058101732812041, 4.43930060989674309234358785788, 4.49125527143314244312390003472, 5.40736753766768044370573974354, 6.12043055137277751861825322723, 6.53745925572367705019854657846, 6.73340307970039576492244038628, 7.62146689440114469921833710882, 7.70561925256738231799922053064, 8.430775478361451180134409531171, 8.541549067508522697176521220742, 9.063032592114854685808495850440, 9.262817905128557648662036611928, 9.846806652314932081194942554342, 9.989660451384092991567192162963
