| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3.30·7-s − 8-s + 9-s − 10-s + 4.58·11-s − 12-s + 0.217·13-s + 3.30·14-s − 15-s + 16-s − 18-s + 7.49·19-s + 20-s + 3.30·21-s − 4.58·22-s − 4.54·23-s + 24-s + 25-s − 0.217·26-s − 27-s − 3.30·28-s − 2.20·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.24·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.38·11-s − 0.288·12-s + 0.0601·13-s + 0.883·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s + 1.71·19-s + 0.223·20-s + 0.721·21-s − 0.976·22-s − 0.946·23-s + 0.204·24-s + 0.200·25-s − 0.0425·26-s − 0.192·27-s − 0.624·28-s − 0.409·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 13 | \( 1 - 0.217T + 13T^{2} \) |
| 19 | \( 1 - 7.49T + 19T^{2} \) |
| 23 | \( 1 + 4.54T + 23T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 + 0.467T + 37T^{2} \) |
| 41 | \( 1 + 4.99T + 41T^{2} \) |
| 43 | \( 1 + 9.86T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 + 8.19T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 5.35T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 + 0.419T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 2.15T + 97T^{2} \) |
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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
−7.10314712130246334454103840440, −6.92012565881639622589544274184, −6.03558771256020247700123729674, −5.74488542419475062243852456313, −4.69518950793150173768741326032, −3.60856561026657262581936070626, −3.18569322317489753936471195333, −1.90566184543880001517454772514, −1.13579753563345024261140256441, 0,
1.13579753563345024261140256441, 1.90566184543880001517454772514, 3.18569322317489753936471195333, 3.60856561026657262581936070626, 4.69518950793150173768741326032, 5.74488542419475062243852456313, 6.03558771256020247700123729674, 6.92012565881639622589544274184, 7.10314712130246334454103840440
