| L(s) = 1 | + 5-s − 3·7-s − 5·11-s − 4·13-s − 16·17-s + 4·19-s − 2·23-s + 5·25-s − 6·29-s + 7·31-s − 3·35-s − 12·37-s + 6·41-s + 2·43-s − 6·47-s + 7·49-s + 10·53-s − 5·55-s + 4·59-s + 8·61-s − 4·65-s + 10·67-s − 16·71-s + 2·73-s + 15·77-s − 16·79-s + 11·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 1.50·11-s − 1.10·13-s − 3.88·17-s + 0.917·19-s − 0.417·23-s + 25-s − 1.11·29-s + 1.25·31-s − 0.507·35-s − 1.97·37-s + 0.937·41-s + 0.304·43-s − 0.875·47-s + 49-s + 1.37·53-s − 0.674·55-s + 0.520·59-s + 1.02·61-s − 0.496·65-s + 1.22·67-s − 1.89·71-s + 0.234·73-s + 1.70·77-s − 1.80·79-s + 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5721181111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5721181111\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 11 T + 38 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−10.66360996990732814275110483766, −10.21363685145160520856649956929, −10.11067171736369095084286729707, −9.329078092379566087095166177136, −9.202038196498745396141718803269, −8.612290333953080605025229388184, −8.383823768535433198358398310430, −7.48874751795885148092459273535, −7.21623668144999844948678972195, −6.69690299726272662083039929900, −6.53814047590166549689885732786, −5.70927099469269085890151193451, −5.40198387116037268220963685447, −4.58913276646234300463257516954, −4.57708591329259953458000160821, −3.61791273554822532334781963248, −2.92665563663078630465048751394, −2.23771027135967991963474408826, −2.22930538120509806762478773937, −0.37614585029604598055325616333,
0.37614585029604598055325616333, 2.22930538120509806762478773937, 2.23771027135967991963474408826, 2.92665563663078630465048751394, 3.61791273554822532334781963248, 4.57708591329259953458000160821, 4.58913276646234300463257516954, 5.40198387116037268220963685447, 5.70927099469269085890151193451, 6.53814047590166549689885732786, 6.69690299726272662083039929900, 7.21623668144999844948678972195, 7.48874751795885148092459273535, 8.383823768535433198358398310430, 8.612290333953080605025229388184, 9.202038196498745396141718803269, 9.329078092379566087095166177136, 10.11067171736369095084286729707, 10.21363685145160520856649956929, 10.66360996990732814275110483766
