| L(s) = 1 | + 6·7-s − 3·9-s − 6·17-s + 16·23-s − 25-s − 2·31-s + 8·41-s − 24·47-s + 13·49-s − 18·63-s + 6·71-s + 20·73-s + 16·79-s + 30·89-s − 12·97-s + 8·103-s − 16·113-s − 36·119-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 18·153-s + 157-s + ⋯ |
| L(s) = 1 | + 2.26·7-s − 9-s − 1.45·17-s + 3.33·23-s − 1/5·25-s − 0.359·31-s + 1.24·41-s − 3.50·47-s + 13/7·49-s − 2.26·63-s + 0.712·71-s + 2.34·73-s + 1.80·79-s + 3.17·89-s − 1.21·97-s + 0.788·103-s − 1.50·113-s − 3.30·119-s − 0.0909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.45·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.249264305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.249264305\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.888161071354656180523733962930, −8.337194469142674370955316014872, −8.059038192245714120733171943119, −7.69184738901398443254624432650, −7.51291834797930461153778785818, −6.73474925958488951880697212055, −6.54587548191389868589729441269, −6.39506969649359133478409167153, −5.44181408048517159346337617031, −5.26413412100482898352992671700, −4.98507176628514266414705853329, −4.69926134624758451802857410658, −4.36875834611106022005202363891, −3.57582313847260523281761488986, −3.30011452594679316933524922633, −2.66040479986794716752042748387, −2.24781884032725154212098605095, −1.76694264643965710521682583200, −1.22327568825959452949790301572, −0.57129034914559401244639489251,
0.57129034914559401244639489251, 1.22327568825959452949790301572, 1.76694264643965710521682583200, 2.24781884032725154212098605095, 2.66040479986794716752042748387, 3.30011452594679316933524922633, 3.57582313847260523281761488986, 4.36875834611106022005202363891, 4.69926134624758451802857410658, 4.98507176628514266414705853329, 5.26413412100482898352992671700, 5.44181408048517159346337617031, 6.39506969649359133478409167153, 6.54587548191389868589729441269, 6.73474925958488951880697212055, 7.51291834797930461153778785818, 7.69184738901398443254624432650, 8.059038192245714120733171943119, 8.337194469142674370955316014872, 8.888161071354656180523733962930
