| L(s) = 1 | − 2·4-s − 4·5-s + 4·7-s − 4·9-s − 2·11-s + 2·13-s + 10·17-s + 4·19-s + 8·20-s + 2·23-s + 4·25-s − 8·28-s − 6·31-s − 16·35-s + 8·36-s − 2·41-s + 4·44-s + 16·45-s + 12·47-s − 4·52-s + 22·53-s + 8·55-s − 4·59-s − 8·61-s − 16·63-s + 8·64-s − 8·65-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s + 1.51·7-s − 4/3·9-s − 0.603·11-s + 0.554·13-s + 2.42·17-s + 0.917·19-s + 1.78·20-s + 0.417·23-s + 4/5·25-s − 1.51·28-s − 1.07·31-s − 2.70·35-s + 4/3·36-s − 0.312·41-s + 0.603·44-s + 2.38·45-s + 1.75·47-s − 0.554·52-s + 3.02·53-s + 1.07·55-s − 0.520·59-s − 1.02·61-s − 2.01·63-s + 64-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.354284875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.354284875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 43 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 22 T + 219 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 120 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24 T + 272 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 215 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 196 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 187 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.358190315418023454215544861975, −8.973916377641076040059756235198, −8.379654631295201290550401329040, −8.320586783088866980162415011314, −7.890441549585051031270931862407, −7.64146939619317867109834206620, −7.46980641098736133819013764924, −6.83563376666240588720467186343, −6.07904758579581001886926511650, −5.54293668377903179308609194785, −5.29571062870753304343400022626, −5.07961553019777875875038953922, −4.58413157866908125296916523784, −3.88745686806713959807894748817, −3.58427047696094590861953772434, −3.45669503450739567071937376540, −2.63969668236081009896094900574, −1.96747799986476865454101501233, −0.917895021923498223993192140613, −0.60206100758918104557594532943,
0.60206100758918104557594532943, 0.917895021923498223993192140613, 1.96747799986476865454101501233, 2.63969668236081009896094900574, 3.45669503450739567071937376540, 3.58427047696094590861953772434, 3.88745686806713959807894748817, 4.58413157866908125296916523784, 5.07961553019777875875038953922, 5.29571062870753304343400022626, 5.54293668377903179308609194785, 6.07904758579581001886926511650, 6.83563376666240588720467186343, 7.46980641098736133819013764924, 7.64146939619317867109834206620, 7.890441549585051031270931862407, 8.320586783088866980162415011314, 8.379654631295201290550401329040, 8.973916377641076040059756235198, 9.358190315418023454215544861975
