| L(s) = 1 | − 9·9-s − 56·11-s − 184·19-s + 276·29-s + 160·31-s + 564·41-s + 110·49-s − 1.19e3·59-s − 436·61-s + 1.71e3·71-s + 64·79-s + 81·81-s + 492·89-s + 504·99-s + 540·101-s − 812·109-s − 310·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.08e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.53·11-s − 2.22·19-s + 1.76·29-s + 0.926·31-s + 2.14·41-s + 0.320·49-s − 2.63·59-s − 0.915·61-s + 2.86·71-s + 0.0911·79-s + 1/9·81-s + 0.585·89-s + 0.511·99-s + 0.532·101-s − 0.713·109-s − 0.232·121-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 − 0.492·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.086115753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.086115753\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 110 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1082 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3102 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24270 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 138 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 80 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 282 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158998 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 150046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 280854 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 596 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 218 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 411430 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 856 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 217970 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1130490 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1075390 T^{2} + p^{6} T^{4} \) |
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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
−10.62982862249468476024716613039, −10.17607902131045893335954717651, −9.586937445872458442276752994396, −9.210712594866233977751292892451, −8.477046759176148689762659150298, −8.444503364934786479309542437310, −7.74488575732266708434251442052, −7.66352902314146214059315340506, −6.78974554956239422098266798307, −6.22862287644665637179862425515, −6.18902719646731206967948829327, −5.42737344006327833307934476871, −4.69333526204984609424030166350, −4.64694166278860126129553492564, −3.90377677473587097742287282586, −3.11820916791642822686111370887, −2.42704248506025690321537381040, −2.35880730505502814543603472359, −1.16896360690496348424190039005, −0.32454214103882052258602910621,
0.32454214103882052258602910621, 1.16896360690496348424190039005, 2.35880730505502814543603472359, 2.42704248506025690321537381040, 3.11820916791642822686111370887, 3.90377677473587097742287282586, 4.64694166278860126129553492564, 4.69333526204984609424030166350, 5.42737344006327833307934476871, 6.18902719646731206967948829327, 6.22862287644665637179862425515, 6.78974554956239422098266798307, 7.66352902314146214059315340506, 7.74488575732266708434251442052, 8.444503364934786479309542437310, 8.477046759176148689762659150298, 9.210712594866233977751292892451, 9.586937445872458442276752994396, 10.17607902131045893335954717651, 10.62982862249468476024716613039
