| L(s) = 1 | + 2-s − 0.0246·3-s + 4-s + 2.47·5-s − 0.0246·6-s + 1.25·7-s + 8-s − 2.99·9-s + 2.47·10-s − 5.17·11-s − 0.0246·12-s − 6.89·13-s + 1.25·14-s − 0.0609·15-s + 16-s − 3.71·17-s − 2.99·18-s + 0.260·19-s + 2.47·20-s − 0.0309·21-s − 5.17·22-s + 6.23·23-s − 0.0246·24-s + 1.13·25-s − 6.89·26-s + 0.147·27-s + 1.25·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0142·3-s + 0.5·4-s + 1.10·5-s − 0.0100·6-s + 0.475·7-s + 0.353·8-s − 0.999·9-s + 0.783·10-s − 1.56·11-s − 0.00710·12-s − 1.91·13-s + 0.336·14-s − 0.0157·15-s + 0.250·16-s − 0.902·17-s − 0.706·18-s + 0.0597·19-s + 0.553·20-s − 0.00675·21-s − 1.10·22-s + 1.30·23-s − 0.00502·24-s + 0.227·25-s − 1.35·26-s + 0.0284·27-s + 0.237·28-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 | \( 1 + 0.0246T + 3T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 - 1.25T + 7T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 + 3.71T + 17T^{2} \) |
| 19 | \( 1 - 0.260T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 + 0.698T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 + 0.636T + 47T^{2} \) |
| 53 | \( 1 - 0.104T + 53T^{2} \) |
| 59 | \( 1 + 6.37T + 59T^{2} \) |
| 61 | \( 1 + 5.45T + 61T^{2} \) |
| 67 | \( 1 + 2.98T + 67T^{2} \) |
| 71 | \( 1 + 9.01T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 0.301T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−7.986229741950360105269756790310, −7.24013344019738865816441622768, −6.50948696805079470286350869515, −5.58201212742431405474341061849, −5.04306647036497657225416983952, −4.74717643521768769874872102669, −3.08824034710417009524775160173, −2.56838793098796349769908363333, −1.88292206770166800035363433922, 0,
1.88292206770166800035363433922, 2.56838793098796349769908363333, 3.08824034710417009524775160173, 4.74717643521768769874872102669, 5.04306647036497657225416983952, 5.58201212742431405474341061849, 6.50948696805079470286350869515, 7.24013344019738865816441622768, 7.986229741950360105269756790310
