| L(s) = 1 | + 40·11-s + 68·17-s − 104·19-s − 242·25-s + 52·41-s − 504·43-s − 486·49-s + 728·59-s − 1.25e3·67-s + 676·73-s + 2.07e3·83-s − 468·89-s − 356·97-s + 2.80e3·107-s − 2.75e3·113-s − 1.46e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.82e3·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.09·11-s + 0.970·17-s − 1.25·19-s − 1.93·25-s + 0.198·41-s − 1.78·43-s − 1.41·49-s + 1.60·59-s − 2.29·67-s + 1.08·73-s + 2.74·83-s − 0.557·89-s − 0.372·97-s + 2.53·107-s − 2.29·113-s − 1.09·121-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 − 1.28·169-s + 0.000439·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 242 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 486 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2826 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 52 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 20462 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8450 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 47414 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 27578 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 252 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 88574 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 166894 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 364 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86838 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 628 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 604430 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 338 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 363350 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1036 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 234 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 178 T + p^{3} T^{2} )^{2} \) |
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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
−8.392566882193190461383723212555, −8.095695799331875232738517375983, −7.71848549060525052244391783534, −7.42765839834505290497398949568, −6.69796375589703776313757486086, −6.62177841644140563394747282250, −5.99918326653607961684501945294, −5.99333389724504484896257737834, −5.18178859276382595488562575001, −4.96216852442439405497745589417, −4.39103685745533526181756431443, −3.89088792794289952681911010412, −3.58464027939436296822874250524, −3.29294098443094349898071781772, −2.33002167082550740335015024882, −2.18278051741095751917592354441, −1.37523135709078101058191235606, −1.16790699232445231879058524156, 0, 0,
1.16790699232445231879058524156, 1.37523135709078101058191235606, 2.18278051741095751917592354441, 2.33002167082550740335015024882, 3.29294098443094349898071781772, 3.58464027939436296822874250524, 3.89088792794289952681911010412, 4.39103685745533526181756431443, 4.96216852442439405497745589417, 5.18178859276382595488562575001, 5.99333389724504484896257737834, 5.99918326653607961684501945294, 6.62177841644140563394747282250, 6.69796375589703776313757486086, 7.42765839834505290497398949568, 7.71848549060525052244391783534, 8.095695799331875232738517375983, 8.392566882193190461383723212555
