| L(s) = 1 | + 2-s + 4-s + 0.484·5-s − 7-s + 8-s + 0.484·10-s + 6.24·11-s − 1.12·13-s − 14-s + 16-s + 3.60·17-s + 4.64·19-s + 0.484·20-s + 6.24·22-s − 23-s − 4.76·25-s − 1.12·26-s − 28-s − 1.76·29-s − 1.60·31-s + 32-s + 3.60·34-s − 0.484·35-s − 3.76·37-s + 4.64·38-s + 0.484·40-s − 2.73·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.216·5-s − 0.377·7-s + 0.353·8-s + 0.153·10-s + 1.88·11-s − 0.311·13-s − 0.267·14-s + 0.250·16-s + 0.875·17-s + 1.06·19-s + 0.108·20-s + 1.33·22-s − 0.208·23-s − 0.952·25-s − 0.220·26-s − 0.188·28-s − 0.327·29-s − 0.289·31-s + 0.176·32-s + 0.619·34-s − 0.0819·35-s − 0.618·37-s + 0.752·38-s + 0.0766·40-s − 0.427·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.391151817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.391151817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.484T + 5T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 - 6.79T + 43T^{2} \) |
| 47 | \( 1 - 6.15T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 8.31T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 3.67T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 5.12T + 97T^{2} \) |
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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
−8.922696457060529854776773361463, −7.78588992678201259666487675827, −7.14496752618210111113455674134, −6.34594650477103543766268246638, −5.75667937413591316470266247780, −4.90509912554503446762542516355, −3.81018803985898986775738239780, −3.44897439722633082491125012755, −2.14721139939268019462067073301, −1.10151722132684880697879359297,
1.10151722132684880697879359297, 2.14721139939268019462067073301, 3.44897439722633082491125012755, 3.81018803985898986775738239780, 4.90509912554503446762542516355, 5.75667937413591316470266247780, 6.34594650477103543766268246638, 7.14496752618210111113455674134, 7.78588992678201259666487675827, 8.922696457060529854776773361463
