| L(s) = 1 | − 0.683·3-s − 4.16·5-s − 7-s − 2.53·9-s + 5.01·11-s + 6.32·13-s + 2.84·15-s + 4.96·17-s + 19-s + 0.683·21-s + 4.32·23-s + 12.3·25-s + 3.78·27-s − 1.20·29-s − 5.69·31-s − 3.42·33-s + 4.16·35-s − 2.79·37-s − 4.32·39-s + 0.164·41-s + 0.796·43-s + 10.5·45-s − 4.68·47-s + 49-s − 3.39·51-s − 4.05·53-s − 20.8·55-s + ⋯ |
| L(s) = 1 | − 0.394·3-s − 1.86·5-s − 0.377·7-s − 0.844·9-s + 1.51·11-s + 1.75·13-s + 0.735·15-s + 1.20·17-s + 0.229·19-s + 0.149·21-s + 0.902·23-s + 2.46·25-s + 0.728·27-s − 0.223·29-s − 1.02·31-s − 0.596·33-s + 0.703·35-s − 0.459·37-s − 0.693·39-s + 0.0256·41-s + 0.121·43-s + 1.57·45-s − 0.683·47-s + 0.142·49-s − 0.475·51-s − 0.556·53-s − 2.81·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223850550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223850550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.683T + 3T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 11 | \( 1 - 5.01T + 11T^{2} \) |
| 13 | \( 1 - 6.32T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 23 | \( 1 - 4.32T + 23T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 + 5.69T + 31T^{2} \) |
| 37 | \( 1 + 2.79T + 37T^{2} \) |
| 41 | \( 1 - 0.164T + 41T^{2} \) |
| 43 | \( 1 - 0.796T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 + 4.05T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 - 1.64T + 61T^{2} \) |
| 67 | \( 1 + 1.03T + 67T^{2} \) |
| 71 | \( 1 - 9.01T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 0.467T + 79T^{2} \) |
| 83 | \( 1 + 4.65T + 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
| 97 | \( 1 + 5.64T + 97T^{2} \) |
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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
−7.83579345786192943984695584159, −7.04171179811398836959951468765, −6.49017833760242067682406754301, −5.78983308678956390963276266188, −4.97050538025636767406596639677, −4.01090025700877096762517552198, −3.49242284823711334910908612793, −3.17863724697596094662929168042, −1.36698627201529995067622304909, −0.62163883261231003078288313739,
0.62163883261231003078288313739, 1.36698627201529995067622304909, 3.17863724697596094662929168042, 3.49242284823711334910908612793, 4.01090025700877096762517552198, 4.97050538025636767406596639677, 5.78983308678956390963276266188, 6.49017833760242067682406754301, 7.04171179811398836959951468765, 7.83579345786192943984695584159
