| L(s) = 1 | + 2.13·2-s + 3.19·3-s + 2.57·4-s − 0.343·5-s + 6.83·6-s − 2.49·7-s + 1.22·8-s + 7.21·9-s − 0.734·10-s + 0.158·11-s + 8.22·12-s − 0.313·13-s − 5.32·14-s − 1.09·15-s − 2.52·16-s − 4.67·17-s + 15.4·18-s − 4.91·19-s − 0.883·20-s − 7.96·21-s + 0.339·22-s + 23-s + 3.92·24-s − 4.88·25-s − 0.670·26-s + 13.4·27-s − 6.41·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.84·3-s + 1.28·4-s − 0.153·5-s + 2.79·6-s − 0.941·7-s + 0.433·8-s + 2.40·9-s − 0.232·10-s + 0.0478·11-s + 2.37·12-s − 0.0869·13-s − 1.42·14-s − 0.283·15-s − 0.630·16-s − 1.13·17-s + 3.63·18-s − 1.12·19-s − 0.197·20-s − 1.73·21-s + 0.0723·22-s + 0.208·23-s + 0.800·24-s − 0.976·25-s − 0.131·26-s + 2.59·27-s − 1.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.007233010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.007233010\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 3 | \( 1 - 3.19T + 3T^{2} \) |
| 5 | \( 1 + 0.343T + 5T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 - 0.158T + 11T^{2} \) |
| 13 | \( 1 + 0.313T + 13T^{2} \) |
| 17 | \( 1 + 4.67T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 31 | \( 1 - 8.73T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 7.50T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 - 5.89T + 53T^{2} \) |
| 59 | \( 1 + 7.19T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 5.79T + 67T^{2} \) |
| 71 | \( 1 + 0.837T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 3.81T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−10.47061236591101767919076580733, −9.457883558867052717572171542371, −8.841404805150236200541563806326, −7.82791313554858158718564189910, −6.82994815705834843947404005996, −6.08682880515593237500531903117, −4.39595349851468571414330101339, −4.02463930546682451106620035243, −2.90003088819194831010911229393, −2.31133758067115764767952379297,
2.31133758067115764767952379297, 2.90003088819194831010911229393, 4.02463930546682451106620035243, 4.39595349851468571414330101339, 6.08682880515593237500531903117, 6.82994815705834843947404005996, 7.82791313554858158718564189910, 8.841404805150236200541563806326, 9.457883558867052717572171542371, 10.47061236591101767919076580733
