| L(s) = 1 | + 1.65·2-s − 1.05·3-s + 0.744·4-s + 1.18·5-s − 1.74·6-s − 3.31·7-s − 2.07·8-s − 1.89·9-s + 1.96·10-s − 2.10·11-s − 0.782·12-s − 0.667·13-s − 5.49·14-s − 1.24·15-s − 4.93·16-s − 5.75·17-s − 3.13·18-s + 7.40·19-s + 0.884·20-s + 3.48·21-s − 3.48·22-s − 23-s + 2.18·24-s − 3.59·25-s − 1.10·26-s + 5.14·27-s − 2.47·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s − 0.606·3-s + 0.372·4-s + 0.530·5-s − 0.710·6-s − 1.25·7-s − 0.735·8-s − 0.631·9-s + 0.621·10-s − 0.633·11-s − 0.225·12-s − 0.185·13-s − 1.46·14-s − 0.322·15-s − 1.23·16-s − 1.39·17-s − 0.739·18-s + 1.69·19-s + 0.197·20-s + 0.761·21-s − 0.742·22-s − 0.208·23-s + 0.446·24-s − 0.718·25-s − 0.216·26-s + 0.990·27-s − 0.466·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.65T + 2T^{2} \) |
| 3 | \( 1 + 1.05T + 3T^{2} \) |
| 5 | \( 1 - 1.18T + 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 + 2.10T + 11T^{2} \) |
| 13 | \( 1 + 0.667T + 13T^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 - 7.40T + 19T^{2} \) |
| 31 | \( 1 - 5.06T + 31T^{2} \) |
| 37 | \( 1 - 2.21T + 37T^{2} \) |
| 41 | \( 1 - 0.348T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 - 1.50T + 61T^{2} \) |
| 67 | \( 1 - 8.31T + 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 4.20T + 83T^{2} \) |
| 89 | \( 1 + 8.05T + 89T^{2} \) |
| 97 | \( 1 + 8.26T + 97T^{2} \) |
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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
−10.00938906369612605026467876498, −9.488735836825453061352140184708, −8.395935451171316032002183085128, −6.91273527702213615814364540363, −6.18617368400733414428955742566, −5.53205885870621399506643923260, −4.72441510426667995806976132032, −3.40321367935097835868040985099, −2.58809866124236602303542578108, 0,
2.58809866124236602303542578108, 3.40321367935097835868040985099, 4.72441510426667995806976132032, 5.53205885870621399506643923260, 6.18617368400733414428955742566, 6.91273527702213615814364540363, 8.395935451171316032002183085128, 9.488735836825453061352140184708, 10.00938906369612605026467876498
