| L(s) = 1 | + 2-s − 0.414·3-s + 4-s − 0.414·6-s − 0.414·7-s + 8-s − 2.82·9-s + 2·11-s − 0.414·12-s + 6.65·13-s − 0.414·14-s + 16-s + 17-s − 2.82·18-s + 0.828·19-s + 0.171·21-s + 2·22-s + 7.65·23-s − 0.414·24-s + 6.65·26-s + 2.41·27-s − 0.414·28-s − 4·29-s + 1.58·31-s + 32-s − 0.828·33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.239·3-s + 0.5·4-s − 0.169·6-s − 0.156·7-s + 0.353·8-s − 0.942·9-s + 0.603·11-s − 0.119·12-s + 1.84·13-s − 0.110·14-s + 0.250·16-s + 0.242·17-s − 0.666·18-s + 0.190·19-s + 0.0374·21-s + 0.426·22-s + 1.59·23-s − 0.0845·24-s + 1.30·26-s + 0.464·27-s − 0.0782·28-s − 0.742·29-s + 0.284·31-s + 0.176·32-s − 0.144·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.317051524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.317051524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 7 | \( 1 + 0.414T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 6.65T + 13T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 1.58T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 - 5.82T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 1.65T + 73T^{2} \) |
| 79 | \( 1 - 1.24T + 79T^{2} \) |
| 83 | \( 1 - 2.34T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.47069240927478387799763706944, −9.198514486617210582380378707185, −8.589187480512094337243592665666, −7.49363843435543192554939825479, −6.36060518200143086884817831643, −5.93135716561353587294568348125, −4.88580149021702740355794221316, −3.72195653013863120453022588324, −2.94480137474653486855259468425, −1.25640618338636972041896897550,
1.25640618338636972041896897550, 2.94480137474653486855259468425, 3.72195653013863120453022588324, 4.88580149021702740355794221316, 5.93135716561353587294568348125, 6.36060518200143086884817831643, 7.49363843435543192554939825479, 8.589187480512094337243592665666, 9.198514486617210582380378707185, 10.47069240927478387799763706944
