L(s) = 1 | + (−0.789 + 1.83i)2-s − 5.33·3-s + (−2.75 − 2.90i)4-s + 2.15i·5-s + (4.21 − 9.79i)6-s + (7.50 − 2.76i)8-s + 19.4·9-s + (−3.96 − 1.70i)10-s + 5.25·11-s + (14.6 + 15.4i)12-s + 21.4i·13-s − 11.5i·15-s + (−0.841 + 15.9i)16-s − 0.926·17-s + (−15.3 + 35.7i)18-s + 5.93·19-s + ⋯ |
L(s) = 1 | + (−0.394 + 0.918i)2-s − 1.77·3-s + (−0.688 − 0.725i)4-s + 0.431i·5-s + (0.701 − 1.63i)6-s + (0.938 − 0.345i)8-s + 2.15·9-s + (−0.396 − 0.170i)10-s + 0.478·11-s + (1.22 + 1.28i)12-s + 1.64i·13-s − 0.766i·15-s + (−0.0525 + 0.998i)16-s − 0.0545·17-s + (−0.852 + 1.98i)18-s + 0.312·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0623630 - 0.349420i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0623630 - 0.349420i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.789 - 1.83i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 5.33T + 9T^{2} \) |
| 5 | \( 1 - 2.15iT - 25T^{2} \) |
| 11 | \( 1 - 5.25T + 121T^{2} \) |
| 13 | \( 1 - 21.4iT - 169T^{2} \) |
| 17 | \( 1 + 0.926T + 289T^{2} \) |
| 19 | \( 1 - 5.93T + 361T^{2} \) |
| 23 | \( 1 + 8.68iT - 529T^{2} \) |
| 29 | \( 1 - 9.42iT - 841T^{2} \) |
| 31 | \( 1 + 34.5iT - 961T^{2} \) |
| 37 | \( 1 - 12.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 43.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 41.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 45.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 53.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 27.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 78.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 74.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 33.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 30.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 72.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 54.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 53.7T + 9.40e3T^{2} \) |
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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
−11.44639012997622995357046333814, −10.72234518461182388846202937528, −9.826208160524757891712081824027, −8.946229769789006687295484043563, −7.44608888930826914701464475753, −6.62604952485922917708092035405, −6.20792104709115171176121722953, −5.01282981360329385985651420417, −4.22722876037187727295900957907, −1.36845183561840673145036291373,
0.27187675342815799947151495075, 1.35850045939903106057005727299, 3.41994523252877762068505357563, 4.82579885726072380319989161756, 5.41480429847835312217406245289, 6.70736217262570970476560909012, 7.84054518841963336430659152234, 8.987877464600847039117035168577, 10.20039237846792239137137818096, 10.53254539105366812903373450581