| L(s) = 1 | + 3-s + 4-s − 2·9-s + 12-s − 8·13-s + 16-s − 6·19-s + 8·25-s − 5·27-s + 10·31-s − 2·36-s + 13·37-s − 8·39-s − 2·43-s + 48-s − 6·49-s − 8·52-s − 6·57-s − 2·61-s + 64-s + 10·67-s − 5·73-s + 8·75-s − 6·76-s − 11·79-s + 81-s + 10·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 2/3·9-s + 0.288·12-s − 2.21·13-s + 1/4·16-s − 1.37·19-s + 8/5·25-s − 0.962·27-s + 1.79·31-s − 1/3·36-s + 2.13·37-s − 1.28·39-s − 0.304·43-s + 0.144·48-s − 6/7·49-s − 1.10·52-s − 0.794·57-s − 0.256·61-s + 1/8·64-s + 1.22·67-s − 0.585·73-s + 0.923·75-s − 0.688·76-s − 1.23·79-s + 1/9·81-s + 1.03·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9677535763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9677535763\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 7 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−12.44178640505011875279644503013, −11.65982210912205254994151809951, −11.23683181354047020207577766874, −10.49550886600262937030096398041, −9.872314871470490477952785695987, −9.430509036835747525566439453077, −8.534981100841526671996842824484, −8.135374084365155596220851926595, −7.39999087139723235793677957867, −6.72100150748654402012240834476, −6.06167660631791350069337618226, −5.00144667887360122896655963437, −4.37969212985630113725934832288, −2.89059835806610854696129231207, −2.44564954087087267979385333309,
2.44564954087087267979385333309, 2.89059835806610854696129231207, 4.37969212985630113725934832288, 5.00144667887360122896655963437, 6.06167660631791350069337618226, 6.72100150748654402012240834476, 7.39999087139723235793677957867, 8.135374084365155596220851926595, 8.534981100841526671996842824484, 9.430509036835747525566439453077, 9.872314871470490477952785695987, 10.49550886600262937030096398041, 11.23683181354047020207577766874, 11.65982210912205254994151809951, 12.44178640505011875279644503013
