| L(s) = 1 | − 3·3-s + 26.7·7-s + 9·9-s − 46.6·11-s − 49.6·13-s − 32.4·17-s − 4.14·19-s − 80.2·21-s − 213.·23-s − 27·27-s − 272.·29-s + 327.·31-s + 139.·33-s + 399.·37-s + 148.·39-s − 42.2·41-s − 468.·43-s + 275.·47-s + 372.·49-s + 97.2·51-s − 158.·53-s + 12.4·57-s + 566.·59-s + 206.·61-s + 240.·63-s + 351.·67-s + 639.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.44·7-s + 0.333·9-s − 1.27·11-s − 1.05·13-s − 0.462·17-s − 0.0500·19-s − 0.833·21-s − 1.93·23-s − 0.192·27-s − 1.74·29-s + 1.89·31-s + 0.737·33-s + 1.77·37-s + 0.610·39-s − 0.161·41-s − 1.66·43-s + 0.853·47-s + 1.08·49-s + 0.266·51-s − 0.409·53-s + 0.0288·57-s + 1.24·59-s + 0.433·61-s + 0.481·63-s + 0.641·67-s + 1.11·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.289499874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.289499874\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 26.7T + 343T^{2} \) |
| 11 | \( 1 + 46.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4.14T + 6.85e3T^{2} \) |
| 23 | \( 1 + 213.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 327.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 399.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 42.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 468.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 158.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 566.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 206.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 351.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 500.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 987.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 172.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 238.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 42.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
| 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
−8.185987957505728424846231394676, −8.021652743844846351893327187879, −7.20917708499928537050140708666, −6.15473167255748977136910077761, −5.33523657534526663995085260851, −4.78677044519512737409724476798, −4.04398690482172205349858348073, −2.50205181367908111227355371551, −1.88395124264112557922632005411, −0.50240137258061716615414321744,
0.50240137258061716615414321744, 1.88395124264112557922632005411, 2.50205181367908111227355371551, 4.04398690482172205349858348073, 4.78677044519512737409724476798, 5.33523657534526663995085260851, 6.15473167255748977136910077761, 7.20917708499928537050140708666, 8.021652743844846351893327187879, 8.185987957505728424846231394676
