| L(s) = 1 | + 2.08·2-s + 0.508·3-s + 2.33·4-s + 5-s + 1.05·6-s − 4.57·7-s + 0.688·8-s − 2.74·9-s + 2.08·10-s − 5.95·11-s + 1.18·12-s + 5.65·13-s − 9.52·14-s + 0.508·15-s − 3.22·16-s − 2.13·17-s − 5.70·18-s − 2.74·19-s + 2.33·20-s − 2.32·21-s − 12.3·22-s − 1.56·23-s + 0.349·24-s + 25-s + 11.7·26-s − 2.91·27-s − 10.6·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.293·3-s + 1.16·4-s + 0.447·5-s + 0.431·6-s − 1.73·7-s + 0.243·8-s − 0.913·9-s + 0.658·10-s − 1.79·11-s + 0.341·12-s + 1.56·13-s − 2.54·14-s + 0.131·15-s − 0.807·16-s − 0.517·17-s − 1.34·18-s − 0.630·19-s + 0.521·20-s − 0.507·21-s − 2.64·22-s − 0.325·23-s + 0.0714·24-s + 0.200·25-s + 2.30·26-s − 0.561·27-s − 2.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 3 | \( 1 - 0.508T + 3T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 + 5.95T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 + 1.56T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 + 7.49T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 6.84T + 61T^{2} \) |
| 67 | \( 1 + 1.58T + 67T^{2} \) |
| 71 | \( 1 - 2.61T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 0.664T + 83T^{2} \) |
| 89 | \( 1 + 4.70T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.782307220124485983729798346156, −7.990228840057926731209273498139, −6.65097214454144233670555133161, −6.18223160334489845607675994875, −5.66814029921559339970308901939, −4.71074089374111619489010720867, −3.58658724628232408240843229163, −3.00984631406177958417924374050, −2.36973635656265337322892266277, 0,
2.36973635656265337322892266277, 3.00984631406177958417924374050, 3.58658724628232408240843229163, 4.71074089374111619489010720867, 5.66814029921559339970308901939, 6.18223160334489845607675994875, 6.65097214454144233670555133161, 7.990228840057926731209273498139, 8.782307220124485983729798346156
