| L(s) = 1 | − 16·4-s + 365·9-s − 534·11-s + 256·16-s + 1.60e3·19-s − 3.55e3·29-s − 5.16e3·31-s − 5.84e3·36-s − 2.38e4·41-s + 8.54e3·44-s − 2.40e3·49-s − 6.18e4·59-s − 3.56e3·61-s − 4.09e3·64-s − 1.03e5·71-s − 2.56e4·76-s + 2.04e5·79-s + 7.41e4·81-s + 6.48e4·89-s − 1.94e5·99-s − 8.26e4·101-s + 2.48e5·109-s + 5.69e4·116-s − 1.08e5·121-s + 8.26e4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.50·9-s − 1.33·11-s + 1/4·16-s + 1.01·19-s − 0.785·29-s − 0.965·31-s − 0.751·36-s − 2.21·41-s + 0.665·44-s − 1/7·49-s − 2.31·59-s − 0.122·61-s − 1/8·64-s − 2.44·71-s − 0.509·76-s + 3.68·79-s + 1.25·81-s + 0.867·89-s − 1.99·99-s − 0.805·101-s + 2.00·109-s + 0.392·116-s − 0.672·121-s + 0.482·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8499403958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8499403958\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 365 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 267 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 438983 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2576545 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 802 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11208586 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 1779 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2584 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 53467130 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 11904 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 293659282 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 169043653 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 60657082 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 30912 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 1780 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2072647510 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 51984 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1871807086 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 102121 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 876408310 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 32400 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 4920655511 T^{2} + p^{10} T^{4} \) |
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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
−11.26507882157582441527594805196, −10.41914184634934101987428519143, −10.03847749962235241470478284008, −9.444253671361105258605931112766, −9.224767498142178671967416007908, −8.584836986860365346770227188951, −7.929230065503577590372147530596, −7.49751665495335295521231117059, −7.38736464034713418818671262855, −6.59377775645135725797102227972, −6.06452569203967957495668016173, −5.28398008307744683293781069694, −5.00080628742235008339539251998, −4.56777304415797017112690431999, −3.67434516714694434045904321654, −3.42691456610732917552542972221, −2.54002011451980800795475318575, −1.72137314228065562029323332847, −1.26327710766254068714589558576, −0.24550440580973704264112226632,
0.24550440580973704264112226632, 1.26327710766254068714589558576, 1.72137314228065562029323332847, 2.54002011451980800795475318575, 3.42691456610732917552542972221, 3.67434516714694434045904321654, 4.56777304415797017112690431999, 5.00080628742235008339539251998, 5.28398008307744683293781069694, 6.06452569203967957495668016173, 6.59377775645135725797102227972, 7.38736464034713418818671262855, 7.49751665495335295521231117059, 7.929230065503577590372147530596, 8.584836986860365346770227188951, 9.224767498142178671967416007908, 9.444253671361105258605931112766, 10.03847749962235241470478284008, 10.41914184634934101987428519143, 11.26507882157582441527594805196
