| L(s) = 1 | + 3-s − 4·7-s − 2·11-s + 4·13-s + 2·17-s − 4·19-s − 4·21-s − 8·23-s − 27-s + 8·29-s − 3·31-s − 2·33-s − 9·37-s + 4·39-s + 12·41-s + 2·43-s − 6·47-s + 9·49-s + 2·51-s + 2·53-s − 4·57-s + 6·59-s + 61-s + 12·67-s − 8·69-s + 20·71-s + 73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s − 0.603·11-s + 1.10·13-s + 0.485·17-s − 0.917·19-s − 0.872·21-s − 1.66·23-s − 0.192·27-s + 1.48·29-s − 0.538·31-s − 0.348·33-s − 1.47·37-s + 0.640·39-s + 1.87·41-s + 0.304·43-s − 0.875·47-s + 9/7·49-s + 0.280·51-s + 0.274·53-s − 0.529·57-s + 0.781·59-s + 0.128·61-s + 1.46·67-s − 0.963·69-s + 2.37·71-s + 0.117·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313800623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313800623\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−9.553141299068998996659063153651, −8.754206415704821322040433115907, −8.452743074243647658344693758958, −8.248553153495679047560846336136, −8.003108149277681927536420272119, −7.17591238316567771519067758180, −7.02192521188085583064093121710, −6.59621176562681413320486846632, −6.09569994569350684666552752670, −5.83711181549240987434063134822, −5.50023426234482004803686528520, −4.87133854066337045746307146690, −4.18367342422753220053866885927, −3.80089271795905957453570817546, −3.68192331937866003151033990781, −2.78664360504738746327124366774, −2.76298182539439411225729599734, −2.05496965683322539606227545092, −1.32468791285325361875660648977, −0.39438523929829473361109998470,
0.39438523929829473361109998470, 1.32468791285325361875660648977, 2.05496965683322539606227545092, 2.76298182539439411225729599734, 2.78664360504738746327124366774, 3.68192331937866003151033990781, 3.80089271795905957453570817546, 4.18367342422753220053866885927, 4.87133854066337045746307146690, 5.50023426234482004803686528520, 5.83711181549240987434063134822, 6.09569994569350684666552752670, 6.59621176562681413320486846632, 7.02192521188085583064093121710, 7.17591238316567771519067758180, 8.003108149277681927536420272119, 8.248553153495679047560846336136, 8.452743074243647658344693758958, 8.754206415704821322040433115907, 9.553141299068998996659063153651
