| L(s) = 1 | + 2-s + 1.60·3-s + 4-s − 3.41·5-s + 1.60·6-s − 0.628·7-s + 8-s − 0.416·9-s − 3.41·10-s + 3.31·11-s + 1.60·12-s + 1.15·13-s − 0.628·14-s − 5.48·15-s + 16-s − 5.49·17-s − 0.416·18-s + 0.266·19-s − 3.41·20-s − 1.00·21-s + 3.31·22-s − 4.04·23-s + 1.60·24-s + 6.65·25-s + 1.15·26-s − 5.49·27-s − 0.628·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.928·3-s + 0.5·4-s − 1.52·5-s + 0.656·6-s − 0.237·7-s + 0.353·8-s − 0.138·9-s − 1.07·10-s + 0.999·11-s + 0.464·12-s + 0.320·13-s − 0.167·14-s − 1.41·15-s + 0.250·16-s − 1.33·17-s − 0.0980·18-s + 0.0611·19-s − 0.763·20-s − 0.220·21-s + 0.706·22-s − 0.843·23-s + 0.328·24-s + 1.33·25-s + 0.226·26-s − 1.05·27-s − 0.118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 1.60T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 0.628T + 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 - 0.266T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 31 | \( 1 + 8.49T + 31T^{2} \) |
| 37 | \( 1 + 1.34T + 37T^{2} \) |
| 41 | \( 1 + 1.71T + 41T^{2} \) |
| 43 | \( 1 + 3.22T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 1.14T + 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + 5.82T + 89T^{2} \) |
| 97 | \( 1 + 6.93T + 97T^{2} \) |
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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
−8.041347620811195738192865970121, −7.39029447265845144158743629212, −6.68525051013427291065634458994, −5.93970874306898601209894600810, −4.72272321430824002548391603765, −4.02793317280728864558072998800, −3.57893459339677985399350142306, −2.81212917721358911112061874798, −1.70849334363831516636825102354, 0,
1.70849334363831516636825102354, 2.81212917721358911112061874798, 3.57893459339677985399350142306, 4.02793317280728864558072998800, 4.72272321430824002548391603765, 5.93970874306898601209894600810, 6.68525051013427291065634458994, 7.39029447265845144158743629212, 8.041347620811195738192865970121
