| L(s) = 1 | + 2-s − 4-s + 4·7-s − 3·8-s − 4·11-s + 2·13-s + 4·14-s − 16-s − 6·17-s − 6·19-s − 4·22-s + 8·23-s + 2·26-s − 4·28-s − 6·29-s + 2·31-s + 5·32-s − 6·34-s + 37-s − 6·38-s + 10·43-s + 4·44-s + 8·46-s − 12·47-s + 9·49-s − 2·52-s + 4·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.51·7-s − 1.06·8-s − 1.20·11-s + 0.554·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s − 1.37·19-s − 0.852·22-s + 1.66·23-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 0.359·31-s + 0.883·32-s − 1.02·34-s + 0.164·37-s − 0.973·38-s + 1.52·43-s + 0.603·44-s + 1.17·46-s − 1.75·47-s + 9/7·49-s − 0.277·52-s + 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.239138167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.239138167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.032339068475156993984294567216, −7.01529406593128118351773260149, −6.32149842192473514286196134924, −5.36540029190826309724596725857, −5.02841281359670039235371162837, −4.39203452662392860353128371286, −3.76016162901980636196475452877, −2.62587192278447247721268715301, −1.99946410062509594965305080475, −0.64982675658537810921550551195,
0.64982675658537810921550551195, 1.99946410062509594965305080475, 2.62587192278447247721268715301, 3.76016162901980636196475452877, 4.39203452662392860353128371286, 5.02841281359670039235371162837, 5.36540029190826309724596725857, 6.32149842192473514286196134924, 7.01529406593128118351773260149, 8.032339068475156993984294567216
