| L(s) = 1 | + 8·2-s + 48·4-s + 256·8-s − 952·11-s + 1.28e3·16-s − 7.61e3·22-s − 7.39e3·23-s + 2.76e3·25-s − 2.78e3·29-s + 6.14e3·32-s + 2.41e4·37-s + 1.94e4·43-s − 4.56e4·44-s − 5.91e4·46-s + 2.21e4·50-s − 8.62e3·53-s − 2.23e4·58-s + 2.86e4·64-s + 4.04e4·67-s − 5.95e4·71-s + 1.93e5·74-s − 6.63e4·79-s + 1.55e5·86-s − 2.43e5·88-s − 3.54e5·92-s + 1.32e5·100-s − 6.89e4·106-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2.37·11-s + 5/4·16-s − 3.35·22-s − 2.91·23-s + 0.885·25-s − 0.615·29-s + 1.06·32-s + 2.90·37-s + 1.60·43-s − 3.55·44-s − 4.12·46-s + 1.25·50-s − 0.421·53-s − 0.870·58-s + 7/8·64-s + 1.10·67-s − 1.40·71-s + 4.10·74-s − 1.19·79-s + 2.26·86-s − 3.35·88-s − 4.37·92-s + 1.32·100-s − 0.596·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.697416213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.697416213\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2766 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 476 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 184958 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2038210 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4545742 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3696 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 1394 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 53548126 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12090 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 77746 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 9724 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 398191362 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4310 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 995172702 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 1602855202 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 20236 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 29792 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4018773970 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 33176 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7865547022 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6162094194 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 16858918114 T^{2} + p^{10} T^{4} \) |
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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
−9.615943258040167851209367975143, −9.448910572047226331441477524930, −8.533886316647964831877404715503, −8.073113077975309939043345265879, −7.76389544854051304723854043807, −7.64263251844939677232673592276, −6.97556720861362857795028986732, −6.41205617707841454877989548869, −5.79215773420580455247651187747, −5.78187075841301616676562829694, −5.33072549818065720668463072414, −4.63058348646369431673294314824, −4.27003863456086106582371012636, −4.02014040205553665362860345220, −3.06413154817449539951363824565, −2.87448636718062448887612130106, −2.22165086031048676654528569414, −2.02638267460216253642694721933, −1.01928466631093931109388511954, −0.29701194573982210511927905876,
0.29701194573982210511927905876, 1.01928466631093931109388511954, 2.02638267460216253642694721933, 2.22165086031048676654528569414, 2.87448636718062448887612130106, 3.06413154817449539951363824565, 4.02014040205553665362860345220, 4.27003863456086106582371012636, 4.63058348646369431673294314824, 5.33072549818065720668463072414, 5.78187075841301616676562829694, 5.79215773420580455247651187747, 6.41205617707841454877989548869, 6.97556720861362857795028986732, 7.64263251844939677232673592276, 7.76389544854051304723854043807, 8.073113077975309939043345265879, 8.533886316647964831877404715503, 9.448910572047226331441477524930, 9.615943258040167851209367975143
