| L(s) = 1 | + 1.31·3-s − 0.257·7-s − 1.26·9-s − 0.645·11-s − 3.54·13-s + 2.56·17-s + 1.71·19-s − 0.338·21-s + 4.60·23-s − 5.61·27-s − 29-s + 8.00·31-s − 0.849·33-s + 10.8·37-s − 4.66·39-s + 11.1·41-s + 4.92·43-s + 2.31·47-s − 6.93·49-s + 3.37·51-s + 0.0599·53-s + 2.25·57-s + 5.46·59-s − 4.95·61-s + 0.325·63-s + 14.5·67-s + 6.06·69-s + ⋯ |
| L(s) = 1 | + 0.760·3-s − 0.0972·7-s − 0.422·9-s − 0.194·11-s − 0.983·13-s + 0.621·17-s + 0.393·19-s − 0.0738·21-s + 0.960·23-s − 1.08·27-s − 0.185·29-s + 1.43·31-s − 0.147·33-s + 1.77·37-s − 0.747·39-s + 1.74·41-s + 0.750·43-s + 0.337·47-s − 0.990·49-s + 0.472·51-s + 0.00822·53-s + 0.298·57-s + 0.711·59-s − 0.634·61-s + 0.0410·63-s + 1.77·67-s + 0.729·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.205068151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.205068151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.31T + 3T^{2} \) |
| 7 | \( 1 + 0.257T + 7T^{2} \) |
| 11 | \( 1 + 0.645T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 31 | \( 1 - 8.00T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 2.31T + 47T^{2} \) |
| 53 | \( 1 - 0.0599T + 53T^{2} \) |
| 59 | \( 1 - 5.46T + 59T^{2} \) |
| 61 | \( 1 + 4.95T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 6.91T + 73T^{2} \) |
| 79 | \( 1 + 6.25T + 79T^{2} \) |
| 83 | \( 1 + 6.67T + 83T^{2} \) |
| 89 | \( 1 + 1.80T + 89T^{2} \) |
| 97 | \( 1 + 2.12T + 97T^{2} \) |
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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
−8.761076172550975174594204465866, −7.903769499404184845451415405905, −7.56325990415928319410742969743, −6.53337531658825065781539587561, −5.66178366986068106002884544571, −4.87370545043982763474454905698, −3.91932073932904701877478495200, −2.86000222450843762662140332705, −2.45830500383311827516666708723, −0.879693462480292468914289494640,
0.879693462480292468914289494640, 2.45830500383311827516666708723, 2.86000222450843762662140332705, 3.91932073932904701877478495200, 4.87370545043982763474454905698, 5.66178366986068106002884544571, 6.53337531658825065781539587561, 7.56325990415928319410742969743, 7.903769499404184845451415405905, 8.761076172550975174594204465866
