| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 2·7-s + 3·8-s + 9-s + 10-s − 3·11-s + 12-s − 2·14-s + 15-s − 16-s + 17-s − 18-s − 5·19-s + 20-s − 2·21-s + 3·22-s + 5·23-s − 3·24-s − 4·25-s − 27-s − 2·28-s − 8·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.534·14-s + 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.436·21-s + 0.639·22-s + 1.04·23-s − 0.612·24-s − 4/5·25-s − 0.192·27-s − 0.377·28-s − 1.48·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
| 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
−14.05198668207111, −13.26262690895112, −12.98077650775668, −12.73422546044961, −11.87402274655559, −11.35969133417189, −11.03338245718110, −10.62958571606673, −9.986620568226519, −9.698174261660556, −8.861320595612926, −8.627518170595817, −7.964510260111879, −7.505086424841740, −7.345832452505721, −6.367844464035168, −5.839092407720137, −5.186417097362344, −4.782339177039644, −4.286131223552292, −3.729730545619676, −2.903038830405907, −2.006811752077592, −1.511814748650793, −0.6248078773783573, 0,
0.6248078773783573, 1.511814748650793, 2.006811752077592, 2.903038830405907, 3.729730545619676, 4.286131223552292, 4.782339177039644, 5.186417097362344, 5.839092407720137, 6.367844464035168, 7.345832452505721, 7.505086424841740, 7.964510260111879, 8.627518170595817, 8.861320595612926, 9.698174261660556, 9.986620568226519, 10.62958571606673, 11.03338245718110, 11.35969133417189, 11.87402274655559, 12.73422546044961, 12.98077650775668, 13.26262690895112, 14.05198668207111
