| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s − 12-s + 4·13-s + 14-s + 2·15-s + 16-s − 18-s + 6·19-s − 2·20-s + 21-s − 23-s + 24-s − 25-s − 4·26-s − 27-s − 28-s − 6·29-s − 2·30-s − 10·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s − 0.447·20-s + 0.218·21-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.365·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−19.88333024170818, −19.05980511654351, −18.47223042384219, −18.00448818577200, −17.04427041462935, −16.42454580332122, −15.70038788352380, −15.51184167929721, −14.27044544033499, −13.41453699093440, −12.42424900747342, −11.89874684663341, −10.99262949385790, −10.69546716789154, −9.423602360911968, −8.967192955246075, −7.651150158035407, −7.423169282829727, −6.191521051528371, −5.480970696189239, −4.055508787912655, −3.252192991329342, −1.484119617876249, 0,
1.484119617876249, 3.252192991329342, 4.055508787912655, 5.480970696189239, 6.191521051528371, 7.423169282829727, 7.651150158035407, 8.967192955246075, 9.423602360911968, 10.69546716789154, 10.99262949385790, 11.89874684663341, 12.42424900747342, 13.41453699093440, 14.27044544033499, 15.51184167929721, 15.70038788352380, 16.42454580332122, 17.04427041462935, 18.00448818577200, 18.47223042384219, 19.05980511654351, 19.88333024170818
