| L(s) = 1 | − 44·11-s + 106·19-s + 44·29-s − 70·31-s + 936·41-s + 325·49-s + 892·59-s + 254·61-s − 72·71-s − 2.73e3·79-s + 288·89-s + 2.88e3·101-s − 1.40e3·109-s − 1.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 1.27·19-s + 0.281·29-s − 0.405·31-s + 3.56·41-s + 0.947·49-s + 1.96·59-s + 0.533·61-s − 0.120·71-s − 3.89·79-s + 0.343·89-s + 2.83·101-s − 1.23·109-s − 0.909·121-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 + 1.99·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.283270983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.283270983\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 325 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4393 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6462 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 53 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20970 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 35 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 28406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 468 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 26747 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 154746 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 446 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 127 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 56195 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 505550 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1368 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 151470 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 144 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 661105 T^{2} + p^{6} T^{4} \) |
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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
−8.978953864563239822186012059711, −8.797085239517514994844994507050, −8.386410561804469857769750618171, −7.64580490535863809511218551226, −7.54008219943278856057609697814, −7.47625665065262454068042498570, −6.71394656566363103196436034771, −6.34742313617586669267331525610, −5.62838586415054428676283680000, −5.59283556204901695296822275329, −5.21112062651992409561914800821, −4.56214095566332762806990356334, −4.10577686294779830313008645946, −3.80897897010173835143947235725, −2.88938623916626015743178619688, −2.82882624961970486927670791969, −2.28651331479010322288551122849, −1.56349116405121224044442575562, −0.840988129822077940071024306653, −0.49358286890047522587309732690,
0.49358286890047522587309732690, 0.840988129822077940071024306653, 1.56349116405121224044442575562, 2.28651331479010322288551122849, 2.82882624961970486927670791969, 2.88938623916626015743178619688, 3.80897897010173835143947235725, 4.10577686294779830313008645946, 4.56214095566332762806990356334, 5.21112062651992409561914800821, 5.59283556204901695296822275329, 5.62838586415054428676283680000, 6.34742313617586669267331525610, 6.71394656566363103196436034771, 7.47625665065262454068042498570, 7.54008219943278856057609697814, 7.64580490535863809511218551226, 8.386410561804469857769750618171, 8.797085239517514994844994507050, 8.978953864563239822186012059711
