L(s) = 1 | − 8.90·2-s − 30.7·3-s + 47.3·4-s + 274.·6-s + 85.8·7-s − 136.·8-s + 704.·9-s − 38.5·11-s − 1.45e3·12-s + 176.·13-s − 764.·14-s − 295.·16-s − 548.·17-s − 6.27e3·18-s − 47.1·19-s − 2.64e3·21-s + 343.·22-s + 3.00e3·23-s + 4.21e3·24-s − 1.57e3·26-s − 1.41e4·27-s + 4.06e3·28-s + 1.08e3·29-s + 3.69e3·31-s + 7.01e3·32-s + 1.18e3·33-s + 4.88e3·34-s + ⋯ |
L(s) = 1 | − 1.57·2-s − 1.97·3-s + 1.48·4-s + 3.10·6-s + 0.661·7-s − 0.756·8-s + 2.89·9-s − 0.0959·11-s − 2.92·12-s + 0.289·13-s − 1.04·14-s − 0.288·16-s − 0.459·17-s − 4.56·18-s − 0.0299·19-s − 1.30·21-s + 0.151·22-s + 1.18·23-s + 1.49·24-s − 0.455·26-s − 3.74·27-s + 0.979·28-s + 0.239·29-s + 0.689·31-s + 1.21·32-s + 0.189·33-s + 0.724·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.6436273620\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6436273620\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
good | 2 | \( 1 + 8.90T + 32T^{2} \) |
| 3 | \( 1 + 30.7T + 243T^{2} \) |
| 7 | \( 1 - 85.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 38.5T + 1.61e5T^{2} \) |
| 13 | \( 1 - 176.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 548.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 47.1T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.00e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.08e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.18e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.99e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 9.71e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.18e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.13e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.21e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.49e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.35e4T + 8.58e9T^{2} \) |
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
−9.344258031132375499794944991543, −8.338124183109893359101443784994, −7.42434642585911917438087728122, −6.80052761210416021042465007005, −5.99952978457030363017982310870, −5.02570422096808302432306540925, −4.24879457295960388981541995911, −2.19881587267941239771941344540, −1.03511510464566213996873534658, −0.65791705438767760219235838213,
0.65791705438767760219235838213, 1.03511510464566213996873534658, 2.19881587267941239771941344540, 4.24879457295960388981541995911, 5.02570422096808302432306540925, 5.99952978457030363017982310870, 6.80052761210416021042465007005, 7.42434642585911917438087728122, 8.338124183109893359101443784994, 9.344258031132375499794944991543