| L(s) = 1 | − 5·3-s − 5·5-s + 27·9-s − 15·11-s − 34·13-s + 25·15-s + 123·17-s + 86·19-s − 54·23-s − 280·27-s − 354·29-s + 212·31-s + 75·33-s − 74·37-s + 170·39-s + 888·41-s − 92·43-s − 135·45-s + 471·47-s − 615·51-s + 180·53-s + 75·55-s − 430·57-s + 144·59-s − 376·61-s + 170·65-s − 356·67-s + ⋯ |
| L(s) = 1 | − 0.962·3-s − 0.447·5-s + 9-s − 0.411·11-s − 0.725·13-s + 0.430·15-s + 1.75·17-s + 1.03·19-s − 0.489·23-s − 1.99·27-s − 2.26·29-s + 1.22·31-s + 0.395·33-s − 0.328·37-s + 0.697·39-s + 3.38·41-s − 0.326·43-s − 0.447·45-s + 1.46·47-s − 1.68·51-s + 0.466·53-s + 0.183·55-s − 0.999·57-s + 0.317·59-s − 0.789·61-s + 0.324·65-s − 0.649·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.197252797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197252797\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 5 T - 2 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15 T - 1106 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 17 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 123 T + 10216 T^{2} - 123 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 86 T + 537 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 54 T - 9251 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 177 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 212 T + 15153 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 p T - 33 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 444 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 46 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 471 T + 118018 T^{2} - 471 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 180 T - 116477 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 144 T - 184643 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 376 T - 85605 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 356 T - 174027 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 818 T + 280107 T^{2} - 818 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 89 T - 485118 T^{2} + 89 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 780 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1140 T + 594631 T^{2} - 1140 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 169 T + p^{3} T^{2} )^{2} \) |
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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
−9.653878959646510431425686621953, −9.573492661330008689593635436760, −9.288354624026992367087186359882, −8.540656192203946908402475461800, −7.76464771848965446265953143492, −7.60128254261588492720532025644, −7.53961193186302476129887403690, −7.06879059015736121877036787552, −6.09576320664532114113630058994, −6.02649579317656860332434438345, −5.46265737014274616727218366797, −5.23654795994774671049217466886, −4.56098364758905647478419462778, −4.08042218372562313237119552194, −3.62623176255686470454202195056, −3.09004760860185324833719476248, −2.29217034027863760794250280753, −1.74953374543297199890480009619, −0.70962602446395108966253659547, −0.63762030867160650430192123642,
0.63762030867160650430192123642, 0.70962602446395108966253659547, 1.74953374543297199890480009619, 2.29217034027863760794250280753, 3.09004760860185324833719476248, 3.62623176255686470454202195056, 4.08042218372562313237119552194, 4.56098364758905647478419462778, 5.23654795994774671049217466886, 5.46265737014274616727218366797, 6.02649579317656860332434438345, 6.09576320664532114113630058994, 7.06879059015736121877036787552, 7.53961193186302476129887403690, 7.60128254261588492720532025644, 7.76464771848965446265953143492, 8.540656192203946908402475461800, 9.288354624026992367087186359882, 9.573492661330008689593635436760, 9.653878959646510431425686621953
