| L(s) = 1 | + 3-s − 2·4-s + 3·5-s + 2·7-s − 2·9-s + 11-s − 2·12-s + 2·13-s + 3·15-s + 4·16-s − 17-s + 2·19-s − 6·20-s + 2·21-s − 3·23-s + 4·25-s − 5·27-s − 4·28-s − 6·29-s − 7·31-s + 33-s + 6·35-s + 4·36-s − 7·37-s + 2·39-s + 12·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.34·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 0.774·15-s + 16-s − 0.242·17-s + 0.458·19-s − 1.34·20-s + 0.436·21-s − 0.625·23-s + 4/5·25-s − 0.962·27-s − 0.755·28-s − 1.11·29-s − 1.25·31-s + 0.174·33-s + 1.01·35-s + 2/3·36-s − 1.15·37-s + 0.320·39-s + 1.87·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.409757951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.409757951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.99782247857233323351447697952, −11.58530592644471232156691500684, −10.41422832511990406315249349300, −9.312607292057116937494029005788, −8.847231817862948898715774476916, −7.75392426065687133350739649603, −5.98785479425231293723674371184, −5.17623765242483563283904939100, −3.64215569583321980752478303848, −1.87664208847796963179892072815,
1.87664208847796963179892072815, 3.64215569583321980752478303848, 5.17623765242483563283904939100, 5.98785479425231293723674371184, 7.75392426065687133350739649603, 8.847231817862948898715774476916, 9.312607292057116937494029005788, 10.41422832511990406315249349300, 11.58530592644471232156691500684, 12.99782247857233323351447697952
