| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 4·13-s + 4·14-s + 5·16-s + 4·17-s + 8·19-s − 4·23-s − 8·26-s − 6·28-s + 4·29-s + 8·31-s − 6·32-s − 8·34-s + 4·37-s − 16·38-s − 12·41-s + 8·43-s + 8·46-s + 8·47-s + 3·49-s + 12·52-s − 12·53-s + 8·56-s − 8·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s + 1.10·13-s + 1.06·14-s + 5/4·16-s + 0.970·17-s + 1.83·19-s − 0.834·23-s − 1.56·26-s − 1.13·28-s + 0.742·29-s + 1.43·31-s − 1.06·32-s − 1.37·34-s + 0.657·37-s − 2.59·38-s − 1.87·41-s + 1.21·43-s + 1.17·46-s + 1.16·47-s + 3/7·49-s + 1.66·52-s − 1.64·53-s + 1.06·56-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8274790315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8274790315\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−11.54672523315593664549491618870, −11.29554923076180718025028882069, −10.55325367837767017951933665498, −10.07728113875637408344071753299, −10.01439033783112463655606599604, −9.485146979364800771182089474166, −8.834744037874522940583574844152, −8.664179296918683319137492382330, −7.86879656006639784858071167283, −7.70971377206101767237854140637, −7.10347793159328129078698097105, −6.53697423851433080849603310003, −5.87882779279032724825489227814, −5.83104606741587686350813742208, −4.80557649010691109432273744906, −3.95932668971741598004652805776, −3.05424496039282091499674828238, −2.97562386654069150960632236205, −1.61740142176884504055417047922, −0.862503666228635144429925468519,
0.862503666228635144429925468519, 1.61740142176884504055417047922, 2.97562386654069150960632236205, 3.05424496039282091499674828238, 3.95932668971741598004652805776, 4.80557649010691109432273744906, 5.83104606741587686350813742208, 5.87882779279032724825489227814, 6.53697423851433080849603310003, 7.10347793159328129078698097105, 7.70971377206101767237854140637, 7.86879656006639784858071167283, 8.664179296918683319137492382330, 8.834744037874522940583574844152, 9.485146979364800771182089474166, 10.01439033783112463655606599604, 10.07728113875637408344071753299, 10.55325367837767017951933665498, 11.29554923076180718025028882069, 11.54672523315593664549491618870
