| L(s) = 1 | − 2·5-s + 4·7-s + 2·11-s + 3·13-s + 8·17-s + 19-s + 8·23-s + 5·25-s + 4·29-s + 6·31-s − 8·35-s + 37-s + 6·41-s − 11·43-s − 12·47-s + 9·49-s − 12·53-s − 4·55-s − 8·59-s − 12·61-s − 6·65-s + 26·67-s + 20·71-s + 11·73-s + 8·77-s − 6·79-s + 2·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 0.603·11-s + 0.832·13-s + 1.94·17-s + 0.229·19-s + 1.66·23-s + 25-s + 0.742·29-s + 1.07·31-s − 1.35·35-s + 0.164·37-s + 0.937·41-s − 1.67·43-s − 1.75·47-s + 9/7·49-s − 1.64·53-s − 0.539·55-s − 1.04·59-s − 1.53·61-s − 0.744·65-s + 3.17·67-s + 2.37·71-s + 1.28·73-s + 0.911·77-s − 0.675·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.789460136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.789460136\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.285702117274318826432183164128, −8.484636550045670076941386799302, −8.331914229154917331808340038655, −8.242043452222187806673227761602, −7.69023672038116709112585594124, −7.53246245241911807888563446648, −6.80859673526925755125187882360, −6.58006057598810795509374823461, −6.22948151303062848260184101696, −5.52621592511868185873439538250, −4.98893842650271006639456741843, −4.96912370270836326265886419669, −4.53217634924388527413796181093, −3.86709123919351988825222384290, −3.38796596737608600423443744943, −3.21140986362217815392390639253, −2.53850821728945269921201718714, −1.61824649618290751723304394337, −1.20498453161441346019652576867, −0.826001396944456149174647900534,
0.826001396944456149174647900534, 1.20498453161441346019652576867, 1.61824649618290751723304394337, 2.53850821728945269921201718714, 3.21140986362217815392390639253, 3.38796596737608600423443744943, 3.86709123919351988825222384290, 4.53217634924388527413796181093, 4.96912370270836326265886419669, 4.98893842650271006639456741843, 5.52621592511868185873439538250, 6.22948151303062848260184101696, 6.58006057598810795509374823461, 6.80859673526925755125187882360, 7.53246245241911807888563446648, 7.69023672038116709112585594124, 8.242043452222187806673227761602, 8.331914229154917331808340038655, 8.484636550045670076941386799302, 9.285702117274318826432183164128
