| L(s) = 1 | − 0.676·3-s + 4.93·7-s − 2.54·9-s + 3.91·11-s − 3.84·13-s + 17-s + 8.03·19-s − 3.33·21-s + 0.561·23-s + 3.74·27-s − 5.52·29-s − 1.46·31-s − 2.64·33-s − 4.03·37-s + 2.59·39-s + 9.38·41-s + 4.17·43-s − 11.5·47-s + 17.3·49-s − 0.676·51-s − 3.34·53-s − 5.43·57-s + 11.5·59-s + 5.35·61-s − 12.5·63-s + 9.08·67-s − 0.379·69-s + ⋯ |
| L(s) = 1 | − 0.390·3-s + 1.86·7-s − 0.847·9-s + 1.18·11-s − 1.06·13-s + 0.242·17-s + 1.84·19-s − 0.728·21-s + 0.117·23-s + 0.721·27-s − 1.02·29-s − 0.262·31-s − 0.460·33-s − 0.663·37-s + 0.416·39-s + 1.46·41-s + 0.636·43-s − 1.68·47-s + 2.48·49-s − 0.0946·51-s − 0.459·53-s − 0.719·57-s + 1.50·59-s + 0.685·61-s − 1.58·63-s + 1.11·67-s − 0.0457·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.868337140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.868337140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 0.676T + 3T^{2} \) |
| 7 | \( 1 - 4.93T + 7T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 13 | \( 1 + 3.84T + 13T^{2} \) |
| 19 | \( 1 - 8.03T + 19T^{2} \) |
| 23 | \( 1 - 0.561T + 23T^{2} \) |
| 29 | \( 1 + 5.52T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 - 9.38T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 3.34T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 67 | \( 1 - 9.08T + 67T^{2} \) |
| 71 | \( 1 + 6.79T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 0.114T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 2.78T + 97T^{2} \) |
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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
−9.262167709522707390539634724666, −8.563820820467819125552382427042, −7.64869909746474493108841012852, −7.18099378408950433234082045867, −5.87083013049904206951521271160, −5.24120265016206787548856356026, −4.58608518498703032599048276971, −3.41907590830254701187057814438, −2.11808414335667399571699961854, −1.03349638331192142224068342010,
1.03349638331192142224068342010, 2.11808414335667399571699961854, 3.41907590830254701187057814438, 4.58608518498703032599048276971, 5.24120265016206787548856356026, 5.87083013049904206951521271160, 7.18099378408950433234082045867, 7.64869909746474493108841012852, 8.563820820467819125552382427042, 9.262167709522707390539634724666
