| L(s) = 1 | + 4·5-s + 4·7-s + 9-s − 11-s − 4·19-s + 2·25-s + 16·35-s − 12·37-s + 20·43-s + 4·45-s − 2·49-s + 28·53-s − 4·55-s + 4·63-s − 4·77-s + 4·79-s + 81-s + 32·83-s − 28·89-s − 16·95-s − 4·97-s − 99-s + 32·107-s − 20·113-s + 121-s − 28·125-s + 127-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.917·19-s + 2/5·25-s + 2.70·35-s − 1.97·37-s + 3.04·43-s + 0.596·45-s − 2/7·49-s + 3.84·53-s − 0.539·55-s + 0.503·63-s − 0.455·77-s + 0.450·79-s + 1/9·81-s + 3.51·83-s − 2.96·89-s − 1.64·95-s − 0.406·97-s − 0.100·99-s + 3.09·107-s − 1.88·113-s + 1/11·121-s − 2.50·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 766656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 766656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.611527442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.611527442\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−8.157325277417324558142664034083, −8.017674089261979349140094913788, −7.26983803440929507346556269687, −7.01506907872700944037082793635, −6.41893627610765861044252902088, −5.78263471331879123598345638239, −5.62860857478680063008223727075, −5.22340952757125710050854485601, −4.60150046530904098097171113225, −4.18020563951979381603420573914, −3.57524581761396247236454027882, −2.55394013627242147469753340699, −2.07502595153963756150674220791, −1.86785039249087723872477729694, −0.965181733530667589348302449434,
0.965181733530667589348302449434, 1.86785039249087723872477729694, 2.07502595153963756150674220791, 2.55394013627242147469753340699, 3.57524581761396247236454027882, 4.18020563951979381603420573914, 4.60150046530904098097171113225, 5.22340952757125710050854485601, 5.62860857478680063008223727075, 5.78263471331879123598345638239, 6.41893627610765861044252902088, 7.01506907872700944037082793635, 7.26983803440929507346556269687, 8.017674089261979349140094913788, 8.157325277417324558142664034083
