| L(s) = 1 | + 2·7-s − 4·11-s + 6·17-s − 4·19-s + 6·23-s − 8·29-s − 2·31-s + 2·37-s + 10·41-s + 4·43-s + 6·47-s − 3·49-s − 6·53-s − 8·59-s + 8·61-s − 67-s − 14·71-s + 6·73-s − 8·77-s + 2·79-s + 12·83-s + 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s + 1.45·17-s − 0.917·19-s + 1.25·23-s − 1.48·29-s − 0.359·31-s + 0.328·37-s + 1.56·41-s + 0.609·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s − 1.04·59-s + 1.02·61-s − 0.122·67-s − 1.66·71-s + 0.702·73-s − 0.911·77-s + 0.225·79-s + 1.31·83-s + 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.04622906099631, −12.66490578843047, −12.35656996278286, −11.63584173484790, −11.17170135452280, −10.79455435879478, −10.51546306607566, −9.901277191383483, −9.254142646068632, −9.060624786474061, −8.297993856542509, −7.861626747245059, −7.543273198456987, −7.204953477233382, −6.360885778938704, −5.802181592863219, −5.490864084886369, −4.896143815647491, −4.535798299324144, −3.786923526081288, −3.302932355214008, −2.612753074631449, −2.191421847985751, −1.439010158897137, −0.8439332596835575, 0,
0.8439332596835575, 1.439010158897137, 2.191421847985751, 2.612753074631449, 3.302932355214008, 3.786923526081288, 4.535798299324144, 4.896143815647491, 5.490864084886369, 5.802181592863219, 6.360885778938704, 7.204953477233382, 7.543273198456987, 7.861626747245059, 8.297993856542509, 9.060624786474061, 9.254142646068632, 9.901277191383483, 10.51546306607566, 10.79455435879478, 11.17170135452280, 11.63584173484790, 12.35656996278286, 12.66490578843047, 13.04622906099631
