| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 2·11-s + 4·13-s + 5·16-s − 8·17-s − 2·19-s + 4·22-s − 2·23-s − 8·26-s + 14·29-s − 2·31-s − 6·32-s + 16·34-s + 20·37-s + 4·38-s − 12·43-s − 6·44-s + 4·46-s − 12·47-s − 4·49-s + 12·52-s − 10·53-s − 28·58-s − 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 0.603·11-s + 1.10·13-s + 5/4·16-s − 1.94·17-s − 0.458·19-s + 0.852·22-s − 0.417·23-s − 1.56·26-s + 2.59·29-s − 0.359·31-s − 1.06·32-s + 2.74·34-s + 3.28·37-s + 0.648·38-s − 1.82·43-s − 0.904·44-s + 0.589·46-s − 1.75·47-s − 4/7·49-s + 1.66·52-s − 1.37·53-s − 3.67·58-s − 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092049198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092049198\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 14 T + 97 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 121 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 113 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 136 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 119 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 139 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
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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
−7.989924368935259494084013187260, −7.938211478461632835787986255338, −7.30755811346491209029838096829, −6.94329357498887064388330507808, −6.42948143787053293633959545237, −6.41067642359163845304717669472, −6.21438757615137251831359660253, −5.83886606385178859449247416572, −4.95371445025594578284804507689, −4.93710449317099884422893134713, −4.44180575882086149079814482534, −4.16883847927963709207677397466, −3.38485342064251915314969414336, −3.25499144425885601531049172616, −2.53792196867044848028835159137, −2.51694515807185686964469623713, −1.75772754313986829444406221493, −1.57476171582157847233334787288, −0.77745784018823468835758999998, −0.40483151281131082281971178295,
0.40483151281131082281971178295, 0.77745784018823468835758999998, 1.57476171582157847233334787288, 1.75772754313986829444406221493, 2.51694515807185686964469623713, 2.53792196867044848028835159137, 3.25499144425885601531049172616, 3.38485342064251915314969414336, 4.16883847927963709207677397466, 4.44180575882086149079814482534, 4.93710449317099884422893134713, 4.95371445025594578284804507689, 5.83886606385178859449247416572, 6.21438757615137251831359660253, 6.41067642359163845304717669472, 6.42948143787053293633959545237, 6.94329357498887064388330507808, 7.30755811346491209029838096829, 7.938211478461632835787986255338, 7.989924368935259494084013187260
