L(s) = 1 | + 1.41i·2-s + (1.89 − 2.32i)3-s − 2.00·4-s + 1.71i·5-s + (3.29 + 2.67i)6-s − 3.39·7-s − 2.82i·8-s + (−1.85 − 8.80i)9-s − 2.42·10-s + 20.1i·11-s + (−3.78 + 4.65i)12-s + 22.6·13-s − 4.79i·14-s + (3.99 + 3.24i)15-s + 4.00·16-s + 19.8i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.630 − 0.776i)3-s − 0.500·4-s + 0.343i·5-s + (0.549 + 0.445i)6-s − 0.484·7-s − 0.353i·8-s + (−0.205 − 0.978i)9-s − 0.242·10-s + 1.83i·11-s + (−0.315 + 0.388i)12-s + 1.74·13-s − 0.342i·14-s + (0.266 + 0.216i)15-s + 0.250·16-s + 1.16i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.78159 + 0.848688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78159 + 0.848688i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-1.89 + 2.32i)T \) |
| 59 | \( 1 - 7.68iT \) |
good | 5 | \( 1 - 1.71iT - 25T^{2} \) |
| 7 | \( 1 + 3.39T + 49T^{2} \) |
| 11 | \( 1 - 20.1iT - 121T^{2} \) |
| 13 | \( 1 - 22.6T + 169T^{2} \) |
| 17 | \( 1 - 19.8iT - 289T^{2} \) |
| 19 | \( 1 - 33.1T + 361T^{2} \) |
| 23 | \( 1 + 17.1iT - 529T^{2} \) |
| 29 | \( 1 + 20.0iT - 841T^{2} \) |
| 31 | \( 1 + 5.04T + 961T^{2} \) |
| 37 | \( 1 - 23.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 13.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 7.24T + 1.84e3T^{2} \) |
| 47 | \( 1 - 64.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 51.6iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 51.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 11.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 128. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 135.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 10.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 12.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 37.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 62.8T + 9.40e3T^{2} \) |
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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
−11.52863412292784632405474979232, −10.18115201203852253514851315245, −9.359678353392823636446011519822, −8.389310084289693103046644484894, −7.51328260020967111189387520408, −6.69753420281237657744029401215, −5.94591636695088371990294236826, −4.28438871586363977393210667377, −3.09744137778646291848111869572, −1.42308606947434085138634658818,
1.02163370497531223346263059273, 3.19113502994651712716963136406, 3.44104446571802936805408118107, 5.01881115661025572436645445289, 5.94676377047828067412432097301, 7.67834740586602441948005751871, 8.902232458257462374862590289836, 9.031008794205859294406071350204, 10.24567288274935438435810349264, 11.14226362830336128300482966075