| L(s) = 1 | − 3-s + 7-s + 9-s + 2·11-s − 2·13-s + 2·19-s − 21-s − 8·23-s − 27-s + 2·29-s − 6·31-s − 2·33-s − 8·37-s + 2·39-s − 10·41-s + 12·47-s + 49-s + 2·53-s − 2·57-s + 2·61-s + 63-s − 4·67-s + 8·69-s + 14·71-s + 2·73-s + 2·77-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.458·19-s − 0.218·21-s − 1.66·23-s − 0.192·27-s + 0.371·29-s − 1.07·31-s − 0.348·33-s − 1.31·37-s + 0.320·39-s − 1.56·41-s + 1.75·47-s + 1/7·49-s + 0.274·53-s − 0.264·57-s + 0.256·61-s + 0.125·63-s − 0.488·67-s + 0.963·69-s + 1.66·71-s + 0.234·73-s + 0.227·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.009203661702398756455309945505, −7.19861115294906712072309669639, −6.64973923815378860120181086470, −5.69030461858780032045688929456, −5.20105383839306855714255126617, −4.23943718625314368336330499723, −3.58317720093392111494719618741, −2.28941705731607088874227804443, −1.39430585166880212945036624152, 0,
1.39430585166880212945036624152, 2.28941705731607088874227804443, 3.58317720093392111494719618741, 4.23943718625314368336330499723, 5.20105383839306855714255126617, 5.69030461858780032045688929456, 6.64973923815378860120181086470, 7.19861115294906712072309669639, 8.009203661702398756455309945505
