| L(s) = 1 | − 2.19·2-s − 3-s + 2.83·4-s + 4.39·5-s + 2.19·6-s − 2.19·7-s − 1.83·8-s + 9-s − 9.66·10-s − 6.39·11-s − 2.83·12-s + 2.39·13-s + 4.83·14-s − 4.39·15-s − 1.63·16-s − 5.30·17-s − 2.19·18-s − 3.63·19-s + 12.4·20-s + 2.19·21-s + 14.0·22-s − 2.72·23-s + 1.83·24-s + 14.3·25-s − 5.27·26-s − 27-s − 6.23·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 0.577·3-s + 1.41·4-s + 1.96·5-s + 0.897·6-s − 0.831·7-s − 0.648·8-s + 0.333·9-s − 3.05·10-s − 1.92·11-s − 0.818·12-s + 0.664·13-s + 1.29·14-s − 1.13·15-s − 0.408·16-s − 1.28·17-s − 0.518·18-s − 0.834·19-s + 2.78·20-s + 0.479·21-s + 2.99·22-s − 0.569·23-s + 0.374·24-s + 2.86·25-s − 1.03·26-s − 0.192·27-s − 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 5 | \( 1 - 4.39T + 5T^{2} \) |
| 7 | \( 1 + 2.19T + 7T^{2} \) |
| 11 | \( 1 + 6.39T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 + 3.63T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 + 2.13T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 - 6.97T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 8.46T + 61T^{2} \) |
| 67 | \( 1 - 2.93T + 67T^{2} \) |
| 71 | \( 1 - 8.13T + 71T^{2} \) |
| 73 | \( 1 + 4.50T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−10.00036664804580760301742020540, −9.402118405239445677020710122364, −8.641465242069553565857388574395, −7.54752468402610472570386863083, −6.39680698516796503761026311742, −6.03309949897770037017741697412, −4.85794872282175382602410703088, −2.62961758106967159577114479873, −1.79741049297650950261876984611, 0,
1.79741049297650950261876984611, 2.62961758106967159577114479873, 4.85794872282175382602410703088, 6.03309949897770037017741697412, 6.39680698516796503761026311742, 7.54752468402610472570386863083, 8.641465242069553565857388574395, 9.402118405239445677020710122364, 10.00036664804580760301742020540
