L(s) = 1 | + 2-s + 1.73·3-s + 4-s − 5-s + 1.73·6-s + 7-s + 8-s − 10-s + 1.73·12-s − 5.73·13-s + 14-s − 1.73·15-s + 16-s − 0.732·17-s − 0.267·19-s − 20-s + 1.73·21-s − 5·23-s + 1.73·24-s + 25-s − 5.73·26-s − 5.19·27-s + 28-s + 8·29-s − 1.73·30-s − 2.73·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.447·5-s + 0.707·6-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.500·12-s − 1.58·13-s + 0.267·14-s − 0.447·15-s + 0.250·16-s − 0.177·17-s − 0.0614·19-s − 0.223·20-s + 0.377·21-s − 1.04·23-s + 0.353·24-s + 0.200·25-s − 1.12·26-s − 1.00·27-s + 0.188·28-s + 1.48·29-s − 0.316·30-s − 0.490·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 19 | \( 1 + 0.267T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 6.19T + 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 + 5.53T + 59T^{2} \) |
| 61 | \( 1 + 2.92T + 61T^{2} \) |
| 67 | \( 1 + 9.66T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 9.12T + 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 - 5T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.57656453329559653650924683028, −6.81566654989916828222344949127, −6.07469671894713726896476165327, −5.08973897601132819678795815427, −4.62261579920303810979909211581, −3.84762583098979055230826690989, −3.02590800743813573302839594135, −2.48505100324600491843778179018, −1.65999264438147676987680905520, 0,
1.65999264438147676987680905520, 2.48505100324600491843778179018, 3.02590800743813573302839594135, 3.84762583098979055230826690989, 4.62261579920303810979909211581, 5.08973897601132819678795815427, 6.07469671894713726896476165327, 6.81566654989916828222344949127, 7.57656453329559653650924683028