L(s) = 1 | + (0.773 − 0.634i)2-s + (0.195 − 0.980i)4-s + (−0.881 − 0.471i)7-s + (−0.471 − 0.881i)8-s + (−0.956 − 0.290i)9-s + (−0.439 − 0.733i)11-s + (−0.980 + 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (−0.805 − 0.288i)22-s + (−0.448 − 0.368i)23-s + (0.995 + 0.0980i)25-s + (−0.634 + 0.773i)28-s + (0.0951 − 0.0238i)29-s + (−0.956 + 0.290i)32-s + ⋯ |
L(s) = 1 | + (0.773 − 0.634i)2-s + (0.195 − 0.980i)4-s + (−0.881 − 0.471i)7-s + (−0.471 − 0.881i)8-s + (−0.956 − 0.290i)9-s + (−0.439 − 0.733i)11-s + (−0.980 + 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (−0.805 − 0.288i)22-s + (−0.448 − 0.368i)23-s + (0.995 + 0.0980i)25-s + (−0.634 + 0.773i)28-s + (0.0951 − 0.0238i)29-s + (−0.956 + 0.290i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.183517500\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.183517500\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.773 + 0.634i)T \) |
| 7 | \( 1 + (0.881 + 0.471i)T \) |
good | 3 | \( 1 + (0.956 + 0.290i)T^{2} \) |
| 5 | \( 1 + (-0.995 - 0.0980i)T^{2} \) |
| 11 | \( 1 + (0.439 + 0.733i)T + (-0.471 + 0.881i)T^{2} \) |
| 13 | \( 1 + (0.0980 + 0.995i)T^{2} \) |
| 17 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 19 | \( 1 + (0.634 - 0.773i)T^{2} \) |
| 23 | \( 1 + (0.448 + 0.368i)T + (0.195 + 0.980i)T^{2} \) |
| 29 | \( 1 + (-0.0951 + 0.0238i)T + (0.881 - 0.471i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (0.346 + 0.968i)T + (-0.773 + 0.634i)T^{2} \) |
| 41 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 43 | \( 1 + (-0.480 + 0.0713i)T + (0.956 - 0.290i)T^{2} \) |
| 47 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 53 | \( 1 + (-1.55 - 0.390i)T + (0.881 + 0.471i)T^{2} \) |
| 59 | \( 1 + (0.0980 - 0.995i)T^{2} \) |
| 61 | \( 1 + (0.290 - 0.956i)T^{2} \) |
| 67 | \( 1 + (-1.58 + 1.17i)T + (0.290 - 0.956i)T^{2} \) |
| 71 | \( 1 + (0.482 + 1.59i)T + (-0.831 + 0.555i)T^{2} \) |
| 73 | \( 1 + (-0.555 + 0.831i)T^{2} \) |
| 79 | \( 1 + (0.162 - 0.108i)T + (0.382 - 0.923i)T^{2} \) |
| 83 | \( 1 + (0.773 + 0.634i)T^{2} \) |
| 89 | \( 1 + (-0.195 + 0.980i)T^{2} \) |
| 97 | \( 1 + (0.707 + 0.707i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.229176945868304617666447163171, −8.592200578802267824365891686951, −7.38700148645444913492413823873, −6.44900102863036765698921947069, −5.87936644159360197327400126334, −5.05251864561019911832005091811, −3.90447242821342077199283816345, −3.21141281091723341269585569975, −2.41175646759658421325313029693, −0.64936663439373444718682567722,
2.36262539271789127577855083848, 3.02701533820577620729569808845, 4.06619994780061033184929063751, 5.15353143501669307552805293841, 5.67071156508148537512672961589, 6.58354516286080557149348379277, 7.21437952110014601601564148245, 8.223515588186714237880264973577, 8.754981464293830046902186934262, 9.697648345233776987276286139366