| L(s) = 1 | − 8·7-s − 9-s − 4·17-s + 16·23-s − 25-s − 20·41-s + 16·47-s + 34·49-s + 8·63-s + 28·73-s + 32·79-s + 81-s − 4·89-s + 4·97-s − 8·103-s + 12·113-s + 32·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s − 128·161-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 1/3·9-s − 0.970·17-s + 3.33·23-s − 1/5·25-s − 3.12·41-s + 2.33·47-s + 34/7·49-s + 1.00·63-s + 3.27·73-s + 3.60·79-s + 1/9·81-s − 0.423·89-s + 0.406·97-s − 0.788·103-s + 1.12·113-s + 2.93·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + 0.0798·157-s − 10.0·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.549801270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549801270\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.703283978798812982518284031266, −8.575601634783396177632168867181, −7.936890754508222546164478020805, −7.36880464654832631127629549157, −6.96123654200229880827507743927, −6.82482739338394676619031540626, −6.50548165426132648462473731846, −6.36634614776945446150254117512, −5.64171542338381521368955388582, −5.44447663328165944894320224664, −4.87754076387487945138627399620, −4.63171332537084281321900687378, −3.67201470556503018140241285121, −3.64794795853322037181115886567, −3.25862163686526172963812835734, −2.82497369172367010098258972885, −2.46997575904783495323288812186, −1.84729202660272661931067424194, −0.66895766369035677396307508918, −0.58681451446960521569042192572,
0.58681451446960521569042192572, 0.66895766369035677396307508918, 1.84729202660272661931067424194, 2.46997575904783495323288812186, 2.82497369172367010098258972885, 3.25862163686526172963812835734, 3.64794795853322037181115886567, 3.67201470556503018140241285121, 4.63171332537084281321900687378, 4.87754076387487945138627399620, 5.44447663328165944894320224664, 5.64171542338381521368955388582, 6.36634614776945446150254117512, 6.50548165426132648462473731846, 6.82482739338394676619031540626, 6.96123654200229880827507743927, 7.36880464654832631127629549157, 7.936890754508222546164478020805, 8.575601634783396177632168867181, 8.703283978798812982518284031266
