| L(s) = 1 | + 2-s + 0.331·3-s + 4-s − 5-s + 0.331·6-s + 4.90·7-s + 8-s − 2.89·9-s − 10-s + 11-s + 0.331·12-s − 6.37·13-s + 4.90·14-s − 0.331·15-s + 16-s + 2.92·17-s − 2.89·18-s − 2.09·19-s − 20-s + 1.62·21-s + 22-s − 3.56·23-s + 0.331·24-s + 25-s − 6.37·26-s − 1.95·27-s + 4.90·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.191·3-s + 0.5·4-s − 0.447·5-s + 0.135·6-s + 1.85·7-s + 0.353·8-s − 0.963·9-s − 0.316·10-s + 0.301·11-s + 0.0956·12-s − 1.76·13-s + 1.31·14-s − 0.0855·15-s + 0.250·16-s + 0.709·17-s − 0.681·18-s − 0.480·19-s − 0.223·20-s + 0.354·21-s + 0.213·22-s − 0.743·23-s + 0.0676·24-s + 0.200·25-s − 1.25·26-s − 0.375·27-s + 0.926·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.592802409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.592802409\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 0.331T + 3T^{2} \) |
| 7 | \( 1 - 4.90T + 7T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 - 2.92T + 17T^{2} \) |
| 19 | \( 1 + 2.09T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 + 2.29T + 29T^{2} \) |
| 31 | \( 1 - 6.50T + 31T^{2} \) |
| 37 | \( 1 - 4.91T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 7.00T + 43T^{2} \) |
| 47 | \( 1 - 9.05T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 - 5.86T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 79 | \( 1 + 4.63T + 79T^{2} \) |
| 83 | \( 1 - 5.01T + 83T^{2} \) |
| 89 | \( 1 - 4.69T + 89T^{2} \) |
| 97 | \( 1 - 5.24T + 97T^{2} \) |
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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
−7.71472227667536978993955022082, −7.37399540422949764176927471186, −6.36510836744840835685770065424, −5.42154880909826649093698890856, −5.08676776625571303434418461143, −4.33926004679774309742167878397, −3.69783345984581454334592169492, −2.47212542690899450638109905988, −2.18328530695494705882096401711, −0.828583178072346764221669005600,
0.828583178072346764221669005600, 2.18328530695494705882096401711, 2.47212542690899450638109905988, 3.69783345984581454334592169492, 4.33926004679774309742167878397, 5.08676776625571303434418461143, 5.42154880909826649093698890856, 6.36510836744840835685770065424, 7.37399540422949764176927471186, 7.71472227667536978993955022082
