| L(s) = 1 | − 4·5-s + 2·11-s − 2·19-s + 11·25-s − 2·29-s − 2·31-s + 13·49-s − 8·55-s + 8·59-s − 14·61-s − 10·71-s − 8·79-s − 14·89-s + 8·95-s + 20·101-s − 20·109-s + 3·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 8·155-s + 157-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.603·11-s − 0.458·19-s + 11/5·25-s − 0.371·29-s − 0.359·31-s + 13/7·49-s − 1.07·55-s + 1.04·59-s − 1.79·61-s − 1.18·71-s − 0.900·79-s − 1.48·89-s + 0.820·95-s + 1.99·101-s − 1.91·109-s + 3/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 0.642·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6666745543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6666745543\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.589848963081571756047539558384, −8.305066141759858781856569625588, −7.892244458135302446353270382093, −7.48508418246425678002759417883, −7.27866468178209409860307934786, −6.91465305387354054356980180088, −6.60852988644062580249078422532, −5.92780326191208509714060750010, −5.84383432462469711236445701703, −5.18854353668630275809302840916, −4.75895185078397810674925630929, −4.36356271220067187678207384617, −4.06550093918151576739486892963, −3.67732394708881986783121408503, −3.38428433933015968593695670739, −2.72704416987439641473355382484, −2.41693139895396653084138123394, −1.54815744790336418410542536875, −1.09345759961013383590024521128, −0.26850257138799107860295856656,
0.26850257138799107860295856656, 1.09345759961013383590024521128, 1.54815744790336418410542536875, 2.41693139895396653084138123394, 2.72704416987439641473355382484, 3.38428433933015968593695670739, 3.67732394708881986783121408503, 4.06550093918151576739486892963, 4.36356271220067187678207384617, 4.75895185078397810674925630929, 5.18854353668630275809302840916, 5.84383432462469711236445701703, 5.92780326191208509714060750010, 6.60852988644062580249078422532, 6.91465305387354054356980180088, 7.27866468178209409860307934786, 7.48508418246425678002759417883, 7.892244458135302446353270382093, 8.305066141759858781856569625588, 8.589848963081571756047539558384
