| L(s) = 1 | − 6·9-s + 40·13-s + 340·25-s + 520·37-s + 1.25e3·49-s + 1.76e3·61-s + 1.64e3·73-s − 693·81-s + 3.08e3·97-s + 4.26e3·109-s − 240·117-s − 2.92e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2/9·9-s + 0.853·13-s + 2.71·25-s + 2.31·37-s + 3.65·49-s + 3.71·61-s + 2.62·73-s − 0.950·81-s + 3.22·97-s + 3.74·109-s − 0.189·117-s − 2.19·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 3.54·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.850582367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.850582367\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 2 p T^{2} + p^{6} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 34 p T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 626 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 1462 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 8546 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 8858 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14926 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 25658 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 9122 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 130 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 122162 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 108554 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 170014 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 380758 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 442 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 60026 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 364178 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 410 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 978818 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 428954 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 703058 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 770 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.573242995142876268444136490823, −8.528029062007044203068344692902, −8.395183939217363798842259757106, −7.79351034233387576704676272556, −7.57014018808288697012341934904, −7.46096774937946149141426312502, −6.84102592397287286603259532981, −6.80386613879901298717059664461, −6.66450691492483079038077022957, −6.08329952447792645773807020609, −5.84532003725618549382750742624, −5.70127132042845253813013182451, −5.28900633405240212186874735122, −4.82179479239538230924550816847, −4.72955359999229265362752703141, −4.33302628779616828306082658065, −3.83823001705736084187568158772, −3.53957109143435531168807794729, −3.39222402185979479726926074389, −2.61882884941525238877980827133, −2.36793381327493036202882384909, −2.26707354860444140372591125598, −1.18664519590623059647119335207, −0.845959511654936657089303113898, −0.69012137169029699312114267735,
0.69012137169029699312114267735, 0.845959511654936657089303113898, 1.18664519590623059647119335207, 2.26707354860444140372591125598, 2.36793381327493036202882384909, 2.61882884941525238877980827133, 3.39222402185979479726926074389, 3.53957109143435531168807794729, 3.83823001705736084187568158772, 4.33302628779616828306082658065, 4.72955359999229265362752703141, 4.82179479239538230924550816847, 5.28900633405240212186874735122, 5.70127132042845253813013182451, 5.84532003725618549382750742624, 6.08329952447792645773807020609, 6.66450691492483079038077022957, 6.80386613879901298717059664461, 6.84102592397287286603259532981, 7.46096774937946149141426312502, 7.57014018808288697012341934904, 7.79351034233387576704676272556, 8.395183939217363798842259757106, 8.528029062007044203068344692902, 8.573242995142876268444136490823
