| L(s) = 1 | + 2-s + 2.30·3-s + 4-s + 5-s + 2.30·6-s + 2.49·7-s + 8-s + 2.31·9-s + 10-s − 5.57·11-s + 2.30·12-s − 1.82·13-s + 2.49·14-s + 2.30·15-s + 16-s + 2.22·17-s + 2.31·18-s + 4.99·19-s + 20-s + 5.75·21-s − 5.57·22-s + 5.08·23-s + 2.30·24-s + 25-s − 1.82·26-s − 1.58·27-s + 2.49·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.33·3-s + 0.5·4-s + 0.447·5-s + 0.940·6-s + 0.943·7-s + 0.353·8-s + 0.770·9-s + 0.316·10-s − 1.67·11-s + 0.665·12-s − 0.507·13-s + 0.667·14-s + 0.595·15-s + 0.250·16-s + 0.540·17-s + 0.545·18-s + 1.14·19-s + 0.223·20-s + 1.25·21-s − 1.18·22-s + 1.05·23-s + 0.470·24-s + 0.200·25-s − 0.358·26-s − 0.304·27-s + 0.471·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.164122108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.164122108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 19 | \( 1 - 4.99T + 19T^{2} \) |
| 23 | \( 1 - 5.08T + 23T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 - 0.739T + 37T^{2} \) |
| 41 | \( 1 - 8.41T + 41T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 53 | \( 1 + 8.07T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 2.22T + 61T^{2} \) |
| 67 | \( 1 + 2.25T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 0.184T + 79T^{2} \) |
| 83 | \( 1 + 3.01T + 83T^{2} \) |
| 89 | \( 1 + 3.64T + 89T^{2} \) |
| 97 | \( 1 + 9.57T + 97T^{2} \) |
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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
−7.907941480050978719383782819962, −7.63934432191384253052447811662, −6.82301413925777377983373036332, −5.56000531377669764997006502754, −5.23754637833273995178897256514, −4.49369950798313244885435165888, −3.44319153806743360620999180808, −2.69096250373973739247435744147, −2.34070184895395488677840851432, −1.17957200844339835327401441841,
1.17957200844339835327401441841, 2.34070184895395488677840851432, 2.69096250373973739247435744147, 3.44319153806743360620999180808, 4.49369950798313244885435165888, 5.23754637833273995178897256514, 5.56000531377669764997006502754, 6.82301413925777377983373036332, 7.63934432191384253052447811662, 7.907941480050978719383782819962
