| L(s) = 1 | + 3-s − 5-s + 3·9-s + 5·11-s − 14·13-s − 15-s − 3·17-s − 2·19-s − 8·23-s + 8·27-s − 10·29-s − 10·31-s + 5·33-s − 4·37-s − 14·39-s + 12·41-s + 4·43-s − 3·45-s − 7·47-s − 3·51-s + 10·53-s − 5·55-s − 2·57-s − 10·59-s − 12·61-s + 14·65-s + 2·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 9-s + 1.50·11-s − 3.88·13-s − 0.258·15-s − 0.727·17-s − 0.458·19-s − 1.66·23-s + 1.53·27-s − 1.85·29-s − 1.79·31-s + 0.870·33-s − 0.657·37-s − 2.24·39-s + 1.87·41-s + 0.609·43-s − 0.447·45-s − 1.02·47-s − 0.420·51-s + 1.37·53-s − 0.674·55-s − 0.264·57-s − 1.30·59-s − 1.53·61-s + 1.73·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5171700785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5171700785\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−9.554865618047112015169978328187, −9.210854637630888485173187769517, −8.681334205111469359054181284010, −8.098882116011955799938026919393, −7.59190676293243162837164728958, −7.47218391235813491148782476949, −7.23991983465693013167243900107, −6.63231020105519545701791562185, −6.54648476028849844763856166234, −5.55095416677555502295516797581, −5.46812048828572835273963905954, −4.71517143913664238175864081230, −4.39279784677336308722538865587, −3.99291241990234586217750288894, −3.84647441648689061973927756693, −2.77881652394566770174618053739, −2.63461398579548136557680576970, −1.79247808136575045115504205402, −1.74186930659965420230859060051, −0.23141901807497023781168673881,
0.23141901807497023781168673881, 1.74186930659965420230859060051, 1.79247808136575045115504205402, 2.63461398579548136557680576970, 2.77881652394566770174618053739, 3.84647441648689061973927756693, 3.99291241990234586217750288894, 4.39279784677336308722538865587, 4.71517143913664238175864081230, 5.46812048828572835273963905954, 5.55095416677555502295516797581, 6.54648476028849844763856166234, 6.63231020105519545701791562185, 7.23991983465693013167243900107, 7.47218391235813491148782476949, 7.59190676293243162837164728958, 8.098882116011955799938026919393, 8.681334205111469359054181284010, 9.210854637630888485173187769517, 9.554865618047112015169978328187
