| L(s) = 1 | − 2-s − 1.06·3-s + 4-s − 0.953·5-s + 1.06·6-s + 4.66·7-s − 8-s − 1.85·9-s + 0.953·10-s − 1.06·12-s + 3.79·13-s − 4.66·14-s + 1.01·15-s + 16-s − 17-s + 1.85·18-s + 0.920·19-s − 0.953·20-s − 4.97·21-s − 7.66·23-s + 1.06·24-s − 4.09·25-s − 3.79·26-s + 5.19·27-s + 4.66·28-s − 2.79·29-s − 1.01·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.616·3-s + 0.5·4-s − 0.426·5-s + 0.436·6-s + 1.76·7-s − 0.353·8-s − 0.619·9-s + 0.301·10-s − 0.308·12-s + 1.05·13-s − 1.24·14-s + 0.262·15-s + 0.250·16-s − 0.242·17-s + 0.438·18-s + 0.211·19-s − 0.213·20-s − 1.08·21-s − 1.59·23-s + 0.218·24-s − 0.818·25-s − 0.743·26-s + 0.998·27-s + 0.880·28-s − 0.518·29-s − 0.185·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.06T + 3T^{2} \) |
| 5 | \( 1 + 0.953T + 5T^{2} \) |
| 7 | \( 1 - 4.66T + 7T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 19 | \( 1 - 0.920T + 19T^{2} \) |
| 23 | \( 1 + 7.66T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 - 0.453T + 37T^{2} \) |
| 41 | \( 1 + 9.57T + 41T^{2} \) |
| 43 | \( 1 - 0.302T + 43T^{2} \) |
| 47 | \( 1 + 4.44T + 47T^{2} \) |
| 53 | \( 1 + 5.60T + 53T^{2} \) |
| 59 | \( 1 + 8.14T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 4.07T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 7.47T + 73T^{2} \) |
| 79 | \( 1 - 3.48T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.053575621139200249959335979994, −7.71944866530087306074779679008, −6.51906002908178947481108374685, −5.93847710826652208378675042943, −5.14087240910061486059716066048, −4.36318843370444127385919309232, −3.40580467126451749629302091786, −2.08949478561904216961699123909, −1.32257817728693584548204184083, 0,
1.32257817728693584548204184083, 2.08949478561904216961699123909, 3.40580467126451749629302091786, 4.36318843370444127385919309232, 5.14087240910061486059716066048, 5.93847710826652208378675042943, 6.51906002908178947481108374685, 7.71944866530087306074779679008, 8.053575621139200249959335979994
