| L(s) = 1 | − 3-s − 5-s + 3·9-s + 5·11-s − 2·13-s + 15-s + 3·17-s − 6·19-s + 6·23-s − 8·27-s − 18·29-s − 5·33-s − 6·37-s + 2·39-s − 16·41-s + 12·43-s − 3·45-s + 3·47-s − 3·51-s + 12·53-s − 5·55-s + 6·57-s + 8·59-s − 4·61-s + 2·65-s + 4·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 9-s + 1.50·11-s − 0.554·13-s + 0.258·15-s + 0.727·17-s − 1.37·19-s + 1.25·23-s − 1.53·27-s − 3.34·29-s − 0.870·33-s − 0.986·37-s + 0.320·39-s − 2.49·41-s + 1.82·43-s − 0.447·45-s + 0.437·47-s − 0.420·51-s + 1.64·53-s − 0.674·55-s + 0.794·57-s + 1.04·59-s − 0.512·61-s + 0.248·65-s + 0.488·67-s − 0.722·69-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
| \(L(1)\) |
\(\approx\) |
\(1.376904034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376904034\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−9.413187686775503848324484192480, −9.199012412406717947207192697616, −8.520327477214815038838283695888, −8.314699177140953543850197130378, −7.56924107095920634290563881298, −7.44919149714935902142360166793, −6.94759679522572795051398381715, −6.70335057879010966034154496457, −6.39045393874449361937492579266, −5.52099986045313030152985312985, −5.46681833125196356230549163569, −5.11621653489701288263683186021, −4.23347632171849850667712800944, −4.05063152095134438788903582666, −3.71729123179919343490785842047, −3.32444755888123149052217274556, −2.25497378302110539873668088568, −1.92266349511086778560839193900, −1.30652332421983998391913855027, −0.46622884866032756500484920644,
0.46622884866032756500484920644, 1.30652332421983998391913855027, 1.92266349511086778560839193900, 2.25497378302110539873668088568, 3.32444755888123149052217274556, 3.71729123179919343490785842047, 4.05063152095134438788903582666, 4.23347632171849850667712800944, 5.11621653489701288263683186021, 5.46681833125196356230549163569, 5.52099986045313030152985312985, 6.39045393874449361937492579266, 6.70335057879010966034154496457, 6.94759679522572795051398381715, 7.44919149714935902142360166793, 7.56924107095920634290563881298, 8.314699177140953543850197130378, 8.520327477214815038838283695888, 9.199012412406717947207192697616, 9.413187686775503848324484192480
