| L(s) = 1 | + 2-s + 0.664·3-s + 4-s − 5-s + 0.664·6-s + 8-s − 2.55·9-s − 10-s − 6.52·11-s + 0.664·12-s + 3.80·13-s − 0.664·15-s + 16-s − 17-s − 2.55·18-s − 6.05·19-s − 20-s − 6.52·22-s − 6.84·23-s + 0.664·24-s + 25-s + 3.80·26-s − 3.69·27-s + 9.62·29-s − 0.664·30-s + 8.49·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.383·3-s + 0.5·4-s − 0.447·5-s + 0.271·6-s + 0.353·8-s − 0.852·9-s − 0.316·10-s − 1.96·11-s + 0.191·12-s + 1.05·13-s − 0.171·15-s + 0.250·16-s − 0.242·17-s − 0.602·18-s − 1.38·19-s − 0.223·20-s − 1.39·22-s − 1.42·23-s + 0.135·24-s + 0.200·25-s + 0.746·26-s − 0.711·27-s + 1.78·29-s − 0.121·30-s + 1.52·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.402351705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.402351705\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.664T + 3T^{2} \) |
| 11 | \( 1 + 6.52T + 11T^{2} \) |
| 13 | \( 1 - 3.80T + 13T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + 3.59T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 2.67T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 0.344T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 3.71T + 67T^{2} \) |
| 71 | \( 1 - 0.0105T + 71T^{2} \) |
| 73 | \( 1 - 7.93T + 73T^{2} \) |
| 79 | \( 1 + 0.721T + 79T^{2} \) |
| 83 | \( 1 + 6.22T + 83T^{2} \) |
| 89 | \( 1 + 6.23T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.995290443031139766806304857136, −7.08437906689349226060501963637, −6.20229180794426555334322837656, −5.79938798748570232534456315164, −4.93253292751157617672067576194, −4.23945775750399630596913048470, −3.53163729943448326798106707679, −2.54089652045214304805175099849, −2.34446802108004190605383207591, −0.63038978248335194283192340667,
0.63038978248335194283192340667, 2.34446802108004190605383207591, 2.54089652045214304805175099849, 3.53163729943448326798106707679, 4.23945775750399630596913048470, 4.93253292751157617672067576194, 5.79938798748570232534456315164, 6.20229180794426555334322837656, 7.08437906689349226060501963637, 7.995290443031139766806304857136
