| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 3.23·7-s + 8-s + 9-s − 10-s + 11-s − 12-s + 1.23·13-s + 3.23·14-s + 15-s + 16-s − 17-s + 18-s − 7.70·19-s − 20-s − 3.23·21-s + 22-s − 5.23·23-s − 24-s + 25-s + 1.23·26-s − 27-s + 3.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.22·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.342·13-s + 0.864·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 1.76·19-s − 0.223·20-s − 0.706·21-s + 0.213·22-s − 1.09·23-s − 0.204·24-s + 0.200·25-s + 0.242·26-s − 0.192·27-s + 0.611·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3.23T + 7T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 19 | \( 1 + 7.70T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.72720140769571260090271352994, −6.92008364183732011081629801422, −6.30743467368498739481516435414, −5.47419669463175265273752797999, −4.90684205222900227494035500445, −4.06183288310022029784484356589, −3.69348550174171394430188070865, −2.15300244560644359040990580256, −1.60966273223528092606398841137, 0,
1.60966273223528092606398841137, 2.15300244560644359040990580256, 3.69348550174171394430188070865, 4.06183288310022029784484356589, 4.90684205222900227494035500445, 5.47419669463175265273752797999, 6.30743467368498739481516435414, 6.92008364183732011081629801422, 7.72720140769571260090271352994
