| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 1.32·7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 6.70·13-s + 1.32·14-s − 15-s + 16-s + 17-s − 18-s − 3.32·19-s + 20-s + 1.32·21-s + 22-s − 4.05·23-s + 24-s + 25-s − 6.70·26-s − 27-s − 1.32·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.501·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 1.86·13-s + 0.354·14-s − 0.258·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 0.762·19-s + 0.223·20-s + 0.289·21-s + 0.213·22-s − 0.845·23-s + 0.204·24-s + 0.200·25-s − 1.31·26-s − 0.192·27-s − 0.250·28-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 | \( 1 + 1.32T + 7T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 23 | \( 1 + 4.05T + 23T^{2} \) |
| 29 | \( 1 + 9.13T + 29T^{2} \) |
| 31 | \( 1 - 9.37T + 31T^{2} \) |
| 37 | \( 1 + 4.97T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 + 9.60T + 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 - 0.185T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 2.40T + 67T^{2} \) |
| 71 | \( 1 - 2.95T + 71T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 - 7.60T + 79T^{2} \) |
| 83 | \( 1 + 6.48T + 83T^{2} \) |
| 89 | \( 1 - 4.76T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| 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
−7.966148293919939968695385409386, −6.96720023259149627440012169261, −6.17665684527435945504969855105, −6.05977349946745963283654729286, −5.05071898525166716244604594132, −3.96315752068079998601762069440, −3.24641724600155060796389946618, −2.06182172534747779152327571110, −1.23625692498825687582583379421, 0,
1.23625692498825687582583379421, 2.06182172534747779152327571110, 3.24641724600155060796389946618, 3.96315752068079998601762069440, 5.05071898525166716244604594132, 6.05977349946745963283654729286, 6.17665684527435945504969855105, 6.96720023259149627440012169261, 7.966148293919939968695385409386
