| L(s) = 1 | − 2.44·2-s + 2.44·3-s + 3.99·4-s + 2.44·5-s − 5.99·6-s − 2.44·7-s − 4.89·8-s + 2.99·9-s − 5.99·10-s − 11-s + 9.79·12-s − 3·13-s + 5.99·14-s + 5.99·15-s + 3.99·16-s − 7·17-s − 7.34·18-s − 4.89·19-s + 9.79·20-s − 5.99·21-s + 2.44·22-s + 23-s − 11.9·24-s + 0.999·25-s + 7.34·26-s − 9.79·28-s + 2.44·29-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.41·3-s + 1.99·4-s + 1.09·5-s − 2.44·6-s − 0.925·7-s − 1.73·8-s + 0.999·9-s − 1.89·10-s − 0.301·11-s + 2.82·12-s − 0.832·13-s + 1.60·14-s + 1.54·15-s + 0.999·16-s − 1.69·17-s − 1.73·18-s − 1.12·19-s + 2.19·20-s − 1.30·21-s + 0.522·22-s + 0.208·23-s − 2.44·24-s + 0.199·25-s + 1.44·26-s − 1.85·28-s + 0.454·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 7.34T + 61T^{2} \) |
| 67 | \( 1 - 9T + 67T^{2} \) |
| 71 | \( 1 + 4.89T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 11T + 97T^{2} \) |
| 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
−9.073381058447805601256365309005, −8.337089079911450708883195215285, −7.60584991633589573479032289791, −6.67863814172410917138263516897, −6.25002369939981202080222924059, −4.66217848583934006532747445107, −3.19392624763796702691461080328, −2.31122467508602671580007165175, −1.92387577974938266647656403220, 0,
1.92387577974938266647656403220, 2.31122467508602671580007165175, 3.19392624763796702691461080328, 4.66217848583934006532747445107, 6.25002369939981202080222924059, 6.67863814172410917138263516897, 7.60584991633589573479032289791, 8.337089079911450708883195215285, 9.073381058447805601256365309005
