| L(s) = 1 | − 2·3-s + 3·7-s + 9-s + 5·11-s + 2·17-s − 5·19-s − 6·21-s + 8·23-s + 4·27-s − 8·31-s − 10·33-s − 9·37-s + 2·41-s + 6·43-s − 7·47-s + 2·49-s − 4·51-s + 13·53-s + 10·57-s + 4·61-s + 3·63-s − 16·69-s − 14·71-s + 10·73-s + 15·77-s − 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.485·17-s − 1.14·19-s − 1.30·21-s + 1.66·23-s + 0.769·27-s − 1.43·31-s − 1.74·33-s − 1.47·37-s + 0.312·41-s + 0.914·43-s − 1.02·47-s + 2/7·49-s − 0.560·51-s + 1.78·53-s + 1.32·57-s + 0.512·61-s + 0.377·63-s − 1.92·69-s − 1.66·71-s + 1.17·73-s + 1.70·77-s − 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.96152050786642, −14.83790173008374, −14.31616091728874, −13.81226683477943, −12.87802688020442, −12.58706188808052, −11.93064099089028, −11.48392786091232, −11.12153455162737, −10.70647003319954, −10.10020493949668, −9.274322055668707, −8.681176285517727, −8.485339421828971, −7.375806415282176, −7.040174947439041, −6.452775394658959, −5.770159478358567, −5.300472109012798, −4.743694927656337, −4.119238740226385, −3.493901982336878, −2.468010591750609, −1.511472939513795, −1.118284734298373, 0,
1.118284734298373, 1.511472939513795, 2.468010591750609, 3.493901982336878, 4.119238740226385, 4.743694927656337, 5.300472109012798, 5.770159478358567, 6.452775394658959, 7.040174947439041, 7.375806415282176, 8.485339421828971, 8.681176285517727, 9.274322055668707, 10.10020493949668, 10.70647003319954, 11.12153455162737, 11.48392786091232, 11.93064099089028, 12.58706188808052, 12.87802688020442, 13.81226683477943, 14.31616091728874, 14.83790173008374, 14.96152050786642
