| L(s) = 1 | + 3-s − 4·7-s + 9-s − 4·11-s + 2·13-s − 6·17-s + 8·19-s − 4·21-s + 27-s + 2·29-s + 8·31-s − 4·33-s + 2·37-s + 2·39-s + 2·41-s − 43-s + 8·47-s + 9·49-s − 6·51-s − 6·53-s + 8·57-s − 12·59-s + 10·61-s − 4·63-s + 4·67-s + 2·73-s + 16·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s − 0.872·21-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.152·43-s + 1.16·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 1.05·57-s − 1.56·59-s + 1.28·61-s − 0.503·63-s + 0.488·67-s + 0.234·73-s + 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.745168056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.745168056\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.31814863267625, −13.78954172434811, −13.44102999143365, −13.16474863114967, −12.52775260196485, −12.09361412189338, −11.34235428384857, −10.81598083526215, −10.24876122646310, −9.741177729578175, −9.358643682628033, −8.835915182441885, −8.136728885822075, −7.750004422453851, −6.976436562886364, −6.634412273350660, −5.983816533916641, −5.386413631050907, −4.677075208086431, −4.016259510336541, −3.339036279071241, −2.738407331827265, −2.512701807798371, −1.316543746588320, −0.4577705328317476,
0.4577705328317476, 1.316543746588320, 2.512701807798371, 2.738407331827265, 3.339036279071241, 4.016259510336541, 4.677075208086431, 5.386413631050907, 5.983816533916641, 6.634412273350660, 6.976436562886364, 7.750004422453851, 8.136728885822075, 8.835915182441885, 9.358643682628033, 9.741177729578175, 10.24876122646310, 10.81598083526215, 11.34235428384857, 12.09361412189338, 12.52775260196485, 13.16474863114967, 13.44102999143365, 13.78954172434811, 14.31814863267625
