| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 3.60·7-s + 8-s + 9-s − 10-s − 2.42·11-s − 12-s + 4.98·13-s − 3.60·14-s + 15-s + 16-s + 18-s + 0.319·19-s − 20-s + 3.60·21-s − 2.42·22-s − 5.84·23-s − 24-s + 25-s + 4.98·26-s − 27-s − 3.60·28-s + 3.05·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 1.36·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.731·11-s − 0.288·12-s + 1.38·13-s − 0.962·14-s + 0.258·15-s + 0.250·16-s + 0.235·18-s + 0.0732·19-s − 0.223·20-s + 0.786·21-s − 0.517·22-s − 1.21·23-s − 0.204·24-s + 0.200·25-s + 0.978·26-s − 0.192·27-s − 0.680·28-s + 0.567·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 2.42T + 11T^{2} \) |
| 13 | \( 1 - 4.98T + 13T^{2} \) |
| 19 | \( 1 - 0.319T + 19T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 - 3.05T + 29T^{2} \) |
| 31 | \( 1 - 6.65T + 31T^{2} \) |
| 37 | \( 1 - 7.24T + 37T^{2} \) |
| 41 | \( 1 + 6.66T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 - 8.01T + 53T^{2} \) |
| 59 | \( 1 + 2.66T + 59T^{2} \) |
| 61 | \( 1 - 6.93T + 61T^{2} \) |
| 67 | \( 1 + 5.90T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 - 3.46T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 1.94T + 97T^{2} \) |
| 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
−7.21292053408716458178174017117, −6.38801884338886864898510875485, −6.20874537893276549071767243088, −5.43869989101344764845346456303, −4.55899952234357087763879665597, −3.87020806625453074913544384795, −3.24858346129994555022555983898, −2.47158340378713831184348909605, −1.14995861875146629031359399499, 0,
1.14995861875146629031359399499, 2.47158340378713831184348909605, 3.24858346129994555022555983898, 3.87020806625453074913544384795, 4.55899952234357087763879665597, 5.43869989101344764845346456303, 6.20874537893276549071767243088, 6.38801884338886864898510875485, 7.21292053408716458178174017117
