| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 5-s + 4·6-s − 4·8-s + 3·9-s − 2·10-s + 6·11-s − 6·12-s − 2·13-s − 2·15-s + 5·16-s − 3·17-s − 6·18-s − 2·19-s + 3·20-s − 12·22-s − 2·23-s + 8·24-s − 5·25-s + 4·26-s − 4·27-s − 8·29-s + 4·30-s − 9·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s − 1.41·8-s + 9-s − 0.632·10-s + 1.80·11-s − 1.73·12-s − 0.554·13-s − 0.516·15-s + 5/4·16-s − 0.727·17-s − 1.41·18-s − 0.458·19-s + 0.670·20-s − 2.55·22-s − 0.417·23-s + 1.63·24-s − 25-s + 0.784·26-s − 0.769·27-s − 1.48·29-s + 0.730·30-s − 1.61·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3910662461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3910662461\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 96 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 126 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 144 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T - 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 182 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 186 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−7.946159228754029699667088160716, −7.84295249717166095573897579099, −7.31236876950751858546155632100, −7.28199452400969424951747682143, −6.71978538037734266834107149620, −6.48137445335870395423981060305, −6.08280406995901827961203328614, −5.84755281523622648776351409902, −5.59547819018845606530905868379, −5.15368050180571316864664662988, −4.40206448072957935548303817768, −4.24077428855369545119480601548, −3.83179066973428519433170235799, −3.51576905874830444891817298948, −2.51513475398525113205969002379, −2.39762913530600876160821952283, −1.80072889305048236619539490981, −1.46327466353418879761095644705, −0.940607411340999308277410648802, −0.25014324993916233053443023079,
0.25014324993916233053443023079, 0.940607411340999308277410648802, 1.46327466353418879761095644705, 1.80072889305048236619539490981, 2.39762913530600876160821952283, 2.51513475398525113205969002379, 3.51576905874830444891817298948, 3.83179066973428519433170235799, 4.24077428855369545119480601548, 4.40206448072957935548303817768, 5.15368050180571316864664662988, 5.59547819018845606530905868379, 5.84755281523622648776351409902, 6.08280406995901827961203328614, 6.48137445335870395423981060305, 6.71978538037734266834107149620, 7.28199452400969424951747682143, 7.31236876950751858546155632100, 7.84295249717166095573897579099, 7.946159228754029699667088160716
