L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s + (−1.36 − 2.36i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 0.464·11-s + (0.499 − 0.866i)12-s + (−3.23 − 5.59i)13-s − 2.73·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−3.86 + 6.69i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.408·6-s + (−0.516 − 0.894i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s − 0.139·11-s + (0.144 − 0.249i)12-s + (−0.896 − 1.55i)13-s − 0.730·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.937 + 1.62i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6031147681\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6031147681\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (4.69 - 3.86i)T \) |
good | 7 | \( 1 + (1.36 + 2.36i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 0.464T + 11T^{2} \) |
| 13 | \( 1 + (3.23 + 5.59i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.86 - 6.69i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.73 - 3i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 5.19T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 41 | \( 1 + (5.36 + 9.29i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 8.39T + 43T^{2} \) |
| 47 | \( 1 - 4.46T + 47T^{2} \) |
| 53 | \( 1 + (-3.36 + 5.83i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.23 + 5.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.83 + 4.90i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.86 + 6.69i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.36 - 5.83i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.83 + 3.16i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.73 - 3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−9.767868970848707423484721307065, −8.543401788650487826442840259605, −8.012276671237530570344669003661, −6.87492421603630711497166676303, −5.73715361428010189329234988800, −4.88996496520490749239706477014, −3.87754848510148034441090222773, −3.33513711040180131429715732365, −1.94051119686729965121213572538, −0.20924664509479009760622936300,
2.25798691326153423309740886718, 2.96302484360948634203723740609, 4.32663463209397410598149125321, 5.17923520161089227997263395432, 6.33605326362960612212773878672, 6.94033243900043366850701057559, 7.49590334255640117331831690759, 8.729957589657839926484136069964, 9.176388968473546584635020324223, 10.00640723478052082804704816828