| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s − 2·9-s − 2·10-s − 11-s + 12-s − 5·13-s + 2·15-s + 16-s + 17-s + 2·18-s − 6·19-s + 2·20-s + 22-s − 24-s − 25-s + 5·26-s − 5·27-s − 6·29-s − 2·30-s − 4·31-s − 32-s − 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 1.38·13-s + 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 1.37·19-s + 0.447·20-s + 0.213·22-s − 0.204·24-s − 1/5·25-s + 0.980·26-s − 0.962·27-s − 1.11·29-s − 0.365·30-s − 0.718·31-s − 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.996710069613788812163657058400, −8.283854810404814822765426999668, −7.57688981483855181683854799897, −6.66604927758438496581469726414, −5.79021686824699936920859385550, −5.01164530936174859299212098766, −3.61174932027046540255643687656, −2.44959014383719530238562200135, −1.97155859884605424765968928907, 0,
1.97155859884605424765968928907, 2.44959014383719530238562200135, 3.61174932027046540255643687656, 5.01164530936174859299212098766, 5.79021686824699936920859385550, 6.66604927758438496581469726414, 7.57688981483855181683854799897, 8.283854810404814822765426999668, 8.996710069613788812163657058400
