| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 3.37·7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 1.37·13-s − 3.37·14-s + 15-s + 16-s − 17-s − 18-s − 3.37·19-s − 20-s − 3.37·21-s + 22-s + 3.37·23-s + 24-s + 25-s + 1.37·26-s − 27-s + 3.37·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.27·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.380·13-s − 0.901·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.773·19-s − 0.223·20-s − 0.735·21-s + 0.213·22-s + 0.703·23-s + 0.204·24-s + 0.200·25-s + 0.269·26-s − 0.192·27-s + 0.637·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 + 2.62T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 - 0.627T + 67T^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 + 7.37T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 9.37T + 97T^{2} \) |
| 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
−7.81288695795899908436022464500, −7.23302495943573180116683692953, −6.50294989804592786989864098744, −5.64862539441396982554805784715, −4.80129743167732526968489091511, −4.36385329639549208884865566510, −3.10589385545724957561097406579, −2.08925950855813743833352800638, −1.20512296867367650905062561195, 0,
1.20512296867367650905062561195, 2.08925950855813743833352800638, 3.10589385545724957561097406579, 4.36385329639549208884865566510, 4.80129743167732526968489091511, 5.64862539441396982554805784715, 6.50294989804592786989864098744, 7.23302495943573180116683692953, 7.81288695795899908436022464500
