| L(s) = 1 | − 2-s + 4·7-s + 8-s − 11-s + 8·13-s − 4·14-s − 16-s − 17-s + 3·19-s + 22-s − 23-s + 5·25-s − 8·26-s + 2·29-s − 6·31-s + 34-s + 3·37-s − 3·38-s + 12·41-s + 2·43-s + 46-s − 47-s + 9·49-s − 5·50-s + 4·56-s − 2·58-s + 7·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.51·7-s + 0.353·8-s − 0.301·11-s + 2.21·13-s − 1.06·14-s − 1/4·16-s − 0.242·17-s + 0.688·19-s + 0.213·22-s − 0.208·23-s + 25-s − 1.56·26-s + 0.371·29-s − 1.07·31-s + 0.171·34-s + 0.493·37-s − 0.486·38-s + 1.87·41-s + 0.304·43-s + 0.147·46-s − 0.145·47-s + 9/7·49-s − 0.707·50-s + 0.534·56-s − 0.262·58-s + 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.408500057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408500057\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−9.565120466471218618658680802780, −9.236724046381275993647095006301, −9.002338443464102858704753666565, −8.545816587868687200787233232187, −8.104830595294451676100375430107, −8.029681256908266097291219361674, −7.44181061763764898260085349480, −7.18304020247927490510410013078, −6.35513047013373148306125061944, −6.23161248907840701119520015073, −5.63451907964691722057356383326, −5.06435612130953063479283479425, −4.88413782263441815320906466744, −4.19118632286208238114467399023, −3.74910637044342925611170918311, −3.34365604398416506472396542466, −2.40532938657690266230165235585, −2.00178093046359585047280429132, −1.07225671076757514254252704793, −0.961111099051834868054928552456,
0.961111099051834868054928552456, 1.07225671076757514254252704793, 2.00178093046359585047280429132, 2.40532938657690266230165235585, 3.34365604398416506472396542466, 3.74910637044342925611170918311, 4.19118632286208238114467399023, 4.88413782263441815320906466744, 5.06435612130953063479283479425, 5.63451907964691722057356383326, 6.23161248907840701119520015073, 6.35513047013373148306125061944, 7.18304020247927490510410013078, 7.44181061763764898260085349480, 8.029681256908266097291219361674, 8.104830595294451676100375430107, 8.545816587868687200787233232187, 9.002338443464102858704753666565, 9.236724046381275993647095006301, 9.565120466471218618658680802780
