| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 3·9-s − 2·11-s − 2·13-s − 4·15-s + 4·17-s − 6·19-s − 4·21-s + 3·25-s − 4·27-s + 4·29-s + 2·31-s + 4·33-s + 4·35-s + 4·37-s + 4·39-s + 4·41-s − 4·43-s + 6·45-s + 3·49-s − 8·51-s − 2·53-s − 4·55-s + 12·57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s − 0.603·11-s − 0.554·13-s − 1.03·15-s + 0.970·17-s − 1.37·19-s − 0.872·21-s + 3/5·25-s − 0.769·27-s + 0.742·29-s + 0.359·31-s + 0.696·33-s + 0.676·35-s + 0.657·37-s + 0.640·39-s + 0.624·41-s − 0.609·43-s + 0.894·45-s + 3/7·49-s − 1.12·51-s − 0.274·53-s − 0.539·55-s + 1.58·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.932925725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.932925725\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.142742959714689961919246396233, −7.82389758625058590301950783520, −7.26319952538656243072310933473, −7.25164144335442151100501031169, −6.46946987729039197030577977881, −6.45383063273886412075943074750, −5.96153218349929408049912422304, −5.83187378769775070657510700050, −5.14732897093301345743613240073, −5.09880520705935399400457081385, −4.67305829474337271056714611224, −4.51102712715630081017310728516, −3.77147443368869694533950532494, −3.54183394192513226373643460769, −2.71480837017893321977452870093, −2.52273585610845977470953891964, −1.84208102599593655266593948468, −1.73919828323421313220491907440, −0.71008858931348487960499533584, −0.68727139888052538684776912316,
0.68727139888052538684776912316, 0.71008858931348487960499533584, 1.73919828323421313220491907440, 1.84208102599593655266593948468, 2.52273585610845977470953891964, 2.71480837017893321977452870093, 3.54183394192513226373643460769, 3.77147443368869694533950532494, 4.51102712715630081017310728516, 4.67305829474337271056714611224, 5.09880520705935399400457081385, 5.14732897093301345743613240073, 5.83187378769775070657510700050, 5.96153218349929408049912422304, 6.45383063273886412075943074750, 6.46946987729039197030577977881, 7.25164144335442151100501031169, 7.26319952538656243072310933473, 7.82389758625058590301950783520, 8.142742959714689961919246396233
