| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 4.47·7-s + 3·8-s + 9-s − 1.23·11-s + 12-s − 5.61·13-s − 4.47·14-s − 16-s − 3.85·17-s − 18-s + 1.23·19-s − 4.47·21-s + 1.23·22-s − 4.47·23-s − 3·24-s + 5.61·26-s − 27-s − 4.47·28-s + 6.61·29-s + 2.76·31-s − 5·32-s + 1.23·33-s + 3.85·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.5·4-s + 0.408·6-s + 1.69·7-s + 1.06·8-s + 0.333·9-s − 0.372·11-s + 0.288·12-s − 1.55·13-s − 1.19·14-s − 0.250·16-s − 0.934·17-s − 0.235·18-s + 0.283·19-s − 0.975·21-s + 0.263·22-s − 0.932·23-s − 0.612·24-s + 1.10·26-s − 0.192·27-s − 0.845·28-s + 1.22·29-s + 0.496·31-s − 0.883·32-s + 0.215·33-s + 0.660·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 - 3.09T + 37T^{2} \) |
| 41 | \( 1 + 3.61T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 - 0.763T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 + 0.763T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 5.38T + 89T^{2} \) |
| 97 | \( 1 + 2.14T + 97T^{2} \) |
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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
−8.730264778061087714549711535863, −8.048192055739203630957445198756, −7.57759296941742729926394613199, −6.62479377385035413489690622074, −5.28833856937473056467156442622, −4.82239830035640573639637618657, −4.24858665355340449188517398080, −2.41606283400103252543369505909, −1.39858431609530956214490230704, 0,
1.39858431609530956214490230704, 2.41606283400103252543369505909, 4.24858665355340449188517398080, 4.82239830035640573639637618657, 5.28833856937473056467156442622, 6.62479377385035413489690622074, 7.57759296941742729926394613199, 8.048192055739203630957445198756, 8.730264778061087714549711535863
