| L(s) = 1 | + 3-s + 1.52·5-s − 1.05·7-s + 9-s + 0.162·11-s + 3.63·13-s + 1.52·15-s + 4.89·17-s + 1.02·19-s − 1.05·21-s − 7.59·23-s − 2.65·25-s + 27-s + 9.59·29-s + 2.68·31-s + 0.162·33-s − 1.61·35-s + 4.12·37-s + 3.63·39-s − 1.08·41-s + 2.77·43-s + 1.52·45-s − 9.50·47-s − 5.89·49-s + 4.89·51-s + 4.80·53-s + 0.247·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.684·5-s − 0.397·7-s + 0.333·9-s + 0.0488·11-s + 1.00·13-s + 0.395·15-s + 1.18·17-s + 0.235·19-s − 0.229·21-s − 1.58·23-s − 0.531·25-s + 0.192·27-s + 1.78·29-s + 0.481·31-s + 0.0282·33-s − 0.272·35-s + 0.677·37-s + 0.582·39-s − 0.168·41-s + 0.423·43-s + 0.228·45-s − 1.38·47-s − 0.841·49-s + 0.685·51-s + 0.660·53-s + 0.0334·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.589912144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.589912144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 - 0.162T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 + 7.59T + 23T^{2} \) |
| 29 | \( 1 - 9.59T + 29T^{2} \) |
| 31 | \( 1 - 2.68T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 + 1.08T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 + 9.50T + 47T^{2} \) |
| 53 | \( 1 - 4.80T + 53T^{2} \) |
| 59 | \( 1 - 8.22T + 59T^{2} \) |
| 61 | \( 1 - 0.510T + 61T^{2} \) |
| 67 | \( 1 + 0.317T + 67T^{2} \) |
| 71 | \( 1 - 5.83T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 5.59T + 79T^{2} \) |
| 83 | \( 1 - 4.98T + 83T^{2} \) |
| 89 | \( 1 + 0.199T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.246519935779262029813901921839, −8.229025250160370782844290559728, −7.911318764001372171379194066517, −6.60249523086355386233009737354, −6.13200762393509435704592563249, −5.21808758995547217275259139520, −4.04365683244958652877943115154, −3.27791306727253261107703386114, −2.26139895169149810908531702003, −1.12911941355696443019815790066,
1.12911941355696443019815790066, 2.26139895169149810908531702003, 3.27791306727253261107703386114, 4.04365683244958652877943115154, 5.21808758995547217275259139520, 6.13200762393509435704592563249, 6.60249523086355386233009737354, 7.911318764001372171379194066517, 8.229025250160370782844290559728, 9.246519935779262029813901921839
