| L(s) = 1 | − 4-s − 2·5-s − 9-s + 16-s + 2·20-s + 3·25-s + 36-s + 4·41-s + 2·45-s − 49-s − 64-s − 2·80-s + 81-s + 4·89-s − 3·100-s − 4·101-s + 4·109-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 4-s − 2·5-s − 9-s + 16-s + 2·20-s + 3·25-s + 36-s + 4·41-s + 2·45-s − 49-s − 64-s − 2·80-s + 81-s + 4·89-s − 3·100-s − 4·101-s + 4·109-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3334573480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3334573480\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−11.79039900547583886584046335973, −11.02650155746776592515764124773, −10.98661769283825681538507651743, −10.47702528357781908294175473677, −9.701440309508268087520475044691, −9.080090020376538106466831649773, −9.074599896973942611869709502385, −8.234637166994617150640613667253, −8.171952976888622737909869378213, −7.48718131269717876772460119877, −7.39997322552566837785584800773, −6.37855064486552610819181081132, −6.01417330079669032641365035135, −5.22305837920699665425129723876, −4.78738509771929236619495438006, −4.20759500455270168954040228116, −3.80714794609498701627206570596, −3.20239717745019535536638769382, −2.57544418646657357988802409667, −0.829579921432508748114186712288,
0.829579921432508748114186712288, 2.57544418646657357988802409667, 3.20239717745019535536638769382, 3.80714794609498701627206570596, 4.20759500455270168954040228116, 4.78738509771929236619495438006, 5.22305837920699665425129723876, 6.01417330079669032641365035135, 6.37855064486552610819181081132, 7.39997322552566837785584800773, 7.48718131269717876772460119877, 8.171952976888622737909869378213, 8.234637166994617150640613667253, 9.074599896973942611869709502385, 9.080090020376538106466831649773, 9.701440309508268087520475044691, 10.47702528357781908294175473677, 10.98661769283825681538507651743, 11.02650155746776592515764124773, 11.79039900547583886584046335973
