| L(s) = 1 | − 2·5-s + 7-s + 2·11-s − 6·13-s + 8·17-s + 19-s + 8·23-s + 5·25-s − 8·29-s − 3·31-s − 2·35-s + 37-s − 12·41-s + 22·43-s + 6·47-s − 6·49-s − 12·53-s − 4·55-s + 4·59-s + 6·61-s + 12·65-s − 13·67-s + 20·71-s + 11·73-s + 2·77-s + 3·79-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.603·11-s − 1.66·13-s + 1.94·17-s + 0.229·19-s + 1.66·23-s + 25-s − 1.48·29-s − 0.538·31-s − 0.338·35-s + 0.164·37-s − 1.87·41-s + 3.35·43-s + 0.875·47-s − 6/7·49-s − 1.64·53-s − 0.539·55-s + 0.520·59-s + 0.768·61-s + 1.48·65-s − 1.58·67-s + 2.37·71-s + 1.28·73-s + 0.227·77-s + 0.337·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.263153378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263153378\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−12.31452891717860794310765759901, −11.83503703506292502175025376957, −11.28791527001990603061071035990, −11.04531449514874741828729448281, −10.42924890499627037924724190065, −9.812583697983873621121745053008, −9.372159469906026931154183920467, −9.042130284714622882543932201253, −8.299391711674754658549173330954, −7.71444404342075596320824127988, −7.30990981320539886578536057919, −7.19961418344622387872940235902, −6.25554651021895590681373542408, −5.48305515935114243213300889725, −5.04613988998220734418384382401, −4.54400292622656144790669268154, −3.60203331204938496924430643433, −3.25553039821128219080739947707, −2.23241068812336342992105072826, −0.965795568859102392831399663720,
0.965795568859102392831399663720, 2.23241068812336342992105072826, 3.25553039821128219080739947707, 3.60203331204938496924430643433, 4.54400292622656144790669268154, 5.04613988998220734418384382401, 5.48305515935114243213300889725, 6.25554651021895590681373542408, 7.19961418344622387872940235902, 7.30990981320539886578536057919, 7.71444404342075596320824127988, 8.299391711674754658549173330954, 9.042130284714622882543932201253, 9.372159469906026931154183920467, 9.812583697983873621121745053008, 10.42924890499627037924724190065, 11.04531449514874741828729448281, 11.28791527001990603061071035990, 11.83503703506292502175025376957, 12.31452891717860794310765759901
