| L(s) = 1 | + (−1.09 − 1.53i)2-s + (−0.235 − 0.971i)3-s + (−0.502 + 1.45i)4-s + (−0.0927 − 1.94i)5-s + (−1.23 + 1.42i)6-s + (0.850 + 2.50i)7-s + (−0.837 + 0.245i)8-s + (−0.888 + 0.458i)9-s + (−2.88 + 2.26i)10-s + (−2.62 + 3.68i)11-s + (1.52 + 0.145i)12-s + (0.355 + 2.47i)13-s + (2.91 − 4.03i)14-s + (−1.87 + 0.549i)15-s + (3.70 + 2.91i)16-s + (−5.23 − 1.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.771 − 1.08i)2-s + (−0.136 − 0.561i)3-s + (−0.251 + 0.725i)4-s + (−0.0414 − 0.870i)5-s + (−0.502 + 0.580i)6-s + (0.321 + 0.946i)7-s + (−0.296 + 0.0869i)8-s + (−0.296 + 0.152i)9-s + (−0.911 + 0.716i)10-s + (−0.790 + 1.11i)11-s + (0.441 + 0.0421i)12-s + (0.0985 + 0.685i)13-s + (0.777 − 1.07i)14-s + (−0.482 + 0.141i)15-s + (0.926 + 0.728i)16-s + (−1.27 − 0.244i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.140819 + 0.0732194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.140819 + 0.0732194i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.235 + 0.971i)T \) |
| 7 | \( 1 + (-0.850 - 2.50i)T \) |
| 23 | \( 1 + (4.10 - 2.47i)T \) |
| good | 2 | \( 1 + (1.09 + 1.53i)T + (-0.654 + 1.89i)T^{2} \) |
| 5 | \( 1 + (0.0927 + 1.94i)T + (-4.97 + 0.475i)T^{2} \) |
| 11 | \( 1 + (2.62 - 3.68i)T + (-3.59 - 10.3i)T^{2} \) |
| 13 | \( 1 + (-0.355 - 2.47i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (5.23 + 1.00i)T + (15.7 + 6.31i)T^{2} \) |
| 19 | \( 1 + (5.79 - 1.11i)T + (17.6 - 7.06i)T^{2} \) |
| 29 | \( 1 + (-1.14 + 1.32i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (4.23 + 4.03i)T + (1.47 + 30.9i)T^{2} \) |
| 37 | \( 1 + (4.15 - 2.14i)T + (21.4 - 30.1i)T^{2} \) |
| 41 | \( 1 + (-6.12 + 3.93i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (2.62 + 0.769i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (2.16 - 3.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.68 + 1.47i)T + (38.3 - 36.5i)T^{2} \) |
| 59 | \( 1 + (-1.11 + 0.875i)T + (13.9 - 57.3i)T^{2} \) |
| 61 | \( 1 + (-2.31 + 9.53i)T + (-54.2 - 27.9i)T^{2} \) |
| 67 | \( 1 + (-11.0 + 1.05i)T + (65.7 - 12.6i)T^{2} \) |
| 71 | \( 1 + (0.430 - 0.941i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.04 - 5.90i)T + (-57.3 - 45.1i)T^{2} \) |
| 79 | \( 1 + (13.5 + 5.40i)T + (57.1 + 54.5i)T^{2} \) |
| 83 | \( 1 + (12.0 + 7.73i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-6.53 + 6.23i)T + (4.23 - 88.8i)T^{2} \) |
| 97 | \( 1 + (4.56 - 2.93i)T + (40.2 - 88.2i)T^{2} \) |
| 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
−11.24646278478172715368897365569, −10.25067339321970184528282092382, −9.259867754803006303826024808708, −8.697807210749029143737230101665, −7.906999566761803172138260178888, −6.53848990640535342112774828038, −5.40068366104843068695328484793, −4.30079398601551678329376479675, −2.33367222310639978404894290405, −1.84022176568237344025793101313,
0.11713717661484256893620258460, 2.87229524517478311496033184099, 4.09333912889677023245791168274, 5.48737243952347377512306722374, 6.47571467717776427475687716612, 7.13397071329703324521728965843, 8.283705668487606718695644503761, 8.646544152807110787685174948148, 10.04803330865267162483751394802, 10.74015242811150745677764641840
