| L(s) = 1 | + 2-s + 2.80·3-s + 4-s + 1.66·5-s + 2.80·6-s + 8-s + 4.88·9-s + 1.66·10-s − 0.605·11-s + 2.80·12-s + 0.466·13-s + 4.67·15-s + 16-s − 17-s + 4.88·18-s − 5.85·19-s + 1.66·20-s − 0.605·22-s + 2.35·23-s + 2.80·24-s − 2.22·25-s + 0.466·26-s + 5.30·27-s − 0.0547·29-s + 4.67·30-s + 10.1·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.62·3-s + 0.5·4-s + 0.744·5-s + 1.14·6-s + 0.353·8-s + 1.62·9-s + 0.526·10-s − 0.182·11-s + 0.810·12-s + 0.129·13-s + 1.20·15-s + 0.250·16-s − 0.242·17-s + 1.15·18-s − 1.34·19-s + 0.372·20-s − 0.129·22-s + 0.491·23-s + 0.573·24-s − 0.445·25-s + 0.0915·26-s + 1.02·27-s − 0.0101·29-s + 0.853·30-s + 1.82·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.152321788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.152321788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 5 | \( 1 - 1.66T + 5T^{2} \) |
| 11 | \( 1 + 0.605T + 11T^{2} \) |
| 13 | \( 1 - 0.466T + 13T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 + 0.0547T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 4.92T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 9.39T + 53T^{2} \) |
| 59 | \( 1 - 4.86T + 59T^{2} \) |
| 61 | \( 1 - 8.39T + 61T^{2} \) |
| 67 | \( 1 - 2.03T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 5.33T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−9.346790018714053271306469666896, −8.421236033956174634217347553376, −8.016889317871795897727907354679, −6.85092635176181363342957737019, −6.27113114948423814139339336200, −5.06479984313558887113042145040, −4.20559250734612847569903911011, −3.27415229263883304577195389695, −2.45634720892264807540344992762, −1.70539637893348936708383394395,
1.70539637893348936708383394395, 2.45634720892264807540344992762, 3.27415229263883304577195389695, 4.20559250734612847569903911011, 5.06479984313558887113042145040, 6.27113114948423814139339336200, 6.85092635176181363342957737019, 8.016889317871795897727907354679, 8.421236033956174634217347553376, 9.346790018714053271306469666896
