| L(s) = 1 | − 2-s − 3.07·3-s + 4-s + 0.455·5-s + 3.07·6-s + 0.869·7-s − 8-s + 6.43·9-s − 0.455·10-s + 3.88·11-s − 3.07·12-s + 1.83·13-s − 0.869·14-s − 1.39·15-s + 16-s − 3.28·17-s − 6.43·18-s + 7.17·19-s + 0.455·20-s − 2.67·21-s − 3.88·22-s − 5.74·23-s + 3.07·24-s − 4.79·25-s − 1.83·26-s − 10.5·27-s + 0.869·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.77·3-s + 0.5·4-s + 0.203·5-s + 1.25·6-s + 0.328·7-s − 0.353·8-s + 2.14·9-s − 0.143·10-s + 1.16·11-s − 0.886·12-s + 0.509·13-s − 0.232·14-s − 0.361·15-s + 0.250·16-s − 0.797·17-s − 1.51·18-s + 1.64·19-s + 0.101·20-s − 0.582·21-s − 0.827·22-s − 1.19·23-s + 0.626·24-s − 0.958·25-s − 0.360·26-s − 2.02·27-s + 0.164·28-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 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 - 0.455T + 5T^{2} \) |
| 7 | \( 1 - 0.869T + 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 19 | \( 1 - 7.17T + 19T^{2} \) |
| 23 | \( 1 + 5.74T + 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 6.99T + 37T^{2} \) |
| 41 | \( 1 + 1.16T + 41T^{2} \) |
| 43 | \( 1 - 4.74T + 43T^{2} \) |
| 47 | \( 1 + 2.06T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 0.819T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 + 7.74T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 6.50T + 73T^{2} \) |
| 79 | \( 1 - 4.56T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 3.11T + 89T^{2} \) |
| 97 | \( 1 + 9.19T + 97T^{2} \) |
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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
−7.982313915290741985620858710576, −7.10086188535934921822111799181, −6.69680176559749698529610354095, −5.74830830164253062888161120759, −5.51310780637641981460572215316, −4.35395481234863097605709144565, −3.62103428169530706444300136684, −1.88107313456311868332079027947, −1.21259545377603984029779976648, 0,
1.21259545377603984029779976648, 1.88107313456311868332079027947, 3.62103428169530706444300136684, 4.35395481234863097605709144565, 5.51310780637641981460572215316, 5.74830830164253062888161120759, 6.69680176559749698529610354095, 7.10086188535934921822111799181, 7.982313915290741985620858710576
