| L(s) = 1 | − 4-s + 4·7-s + 2·13-s − 3·16-s + 4·19-s − 10·25-s − 4·28-s + 4·31-s + 4·37-s + 16·43-s − 2·49-s − 2·52-s − 20·61-s + 7·64-s + 28·67-s − 20·73-s − 4·76-s − 8·79-s + 8·91-s − 20·97-s + 10·100-s − 8·103-s − 20·109-s − 12·112-s − 10·121-s − 4·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.51·7-s + 0.554·13-s − 3/4·16-s + 0.917·19-s − 2·25-s − 0.755·28-s + 0.718·31-s + 0.657·37-s + 2.43·43-s − 2/7·49-s − 0.277·52-s − 2.56·61-s + 7/8·64-s + 3.42·67-s − 2.34·73-s − 0.458·76-s − 0.900·79-s + 0.838·91-s − 2.03·97-s + 100-s − 0.788·103-s − 1.91·109-s − 1.13·112-s − 0.909·121-s − 0.359·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098456692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098456692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 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
−13.78077984813502739949344616735, −13.46817795999654429060682580786, −12.84158329362133660754913360080, −12.14597944474043241962403385795, −11.59558476998937132334248657801, −11.31185930034328724256934016158, −10.82879460641919519696328322075, −10.11191089374798319109999442731, −9.333019073369794308499960380212, −9.225081096982506476833589253554, −8.169496008074871911723438887652, −8.035875394685275233006386448235, −7.43939891686361858773967966201, −6.55613724753903805714270497984, −5.76356334736852262895132650365, −5.24512011077062206601458015712, −4.37792935628374435376605097839, −4.03630049463173406978464715852, −2.69057717548439606251880318396, −1.51141997204695526390325899979,
1.51141997204695526390325899979, 2.69057717548439606251880318396, 4.03630049463173406978464715852, 4.37792935628374435376605097839, 5.24512011077062206601458015712, 5.76356334736852262895132650365, 6.55613724753903805714270497984, 7.43939891686361858773967966201, 8.035875394685275233006386448235, 8.169496008074871911723438887652, 9.225081096982506476833589253554, 9.333019073369794308499960380212, 10.11191089374798319109999442731, 10.82879460641919519696328322075, 11.31185930034328724256934016158, 11.59558476998937132334248657801, 12.14597944474043241962403385795, 12.84158329362133660754913360080, 13.46817795999654429060682580786, 13.78077984813502739949344616735
