L(s) = 1 | + (0.0490 − 0.998i)2-s + (−0.995 − 0.0980i)4-s + (0.595 − 0.803i)7-s + (−0.146 + 0.989i)8-s + (−0.242 − 0.970i)9-s + (1.19 − 1.38i)11-s + (−0.773 − 0.634i)14-s + (0.980 + 0.195i)16-s + (−0.980 + 0.195i)18-s + (−1.32 − 1.26i)22-s + (0.0841 + 1.71i)23-s + (0.427 − 0.903i)25-s + (−0.671 + 0.740i)28-s + (0.416 − 0.137i)29-s + (0.242 − 0.970i)32-s + ⋯ |
L(s) = 1 | + (0.0490 − 0.998i)2-s + (−0.995 − 0.0980i)4-s + (0.595 − 0.803i)7-s + (−0.146 + 0.989i)8-s + (−0.242 − 0.970i)9-s + (1.19 − 1.38i)11-s + (−0.773 − 0.634i)14-s + (0.980 + 0.195i)16-s + (−0.980 + 0.195i)18-s + (−1.32 − 1.26i)22-s + (0.0841 + 1.71i)23-s + (0.427 − 0.903i)25-s + (−0.671 + 0.740i)28-s + (0.416 − 0.137i)29-s + (0.242 − 0.970i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.274902723\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.274902723\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0490 + 0.998i)T \) |
| 7 | \( 1 + (-0.595 + 0.803i)T \) |
good | 3 | \( 1 + (0.242 + 0.970i)T^{2} \) |
| 5 | \( 1 + (-0.427 + 0.903i)T^{2} \) |
| 11 | \( 1 + (-1.19 + 1.38i)T + (-0.146 - 0.989i)T^{2} \) |
| 13 | \( 1 + (-0.941 - 0.336i)T^{2} \) |
| 17 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 19 | \( 1 + (-0.671 + 0.740i)T^{2} \) |
| 23 | \( 1 + (-0.0841 - 1.71i)T + (-0.995 + 0.0980i)T^{2} \) |
| 29 | \( 1 + (-0.416 + 0.137i)T + (0.803 - 0.595i)T^{2} \) |
| 31 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 37 | \( 1 + (0.0320 - 1.30i)T + (-0.998 - 0.0490i)T^{2} \) |
| 41 | \( 1 + (0.634 - 0.773i)T^{2} \) |
| 43 | \( 1 + (0.892 + 0.110i)T + (0.970 + 0.242i)T^{2} \) |
| 47 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 53 | \( 1 + (0.936 + 0.309i)T + (0.803 + 0.595i)T^{2} \) |
| 59 | \( 1 + (0.941 - 0.336i)T^{2} \) |
| 61 | \( 1 + (-0.857 - 0.514i)T^{2} \) |
| 67 | \( 1 + (0.120 + 0.213i)T + (-0.514 + 0.857i)T^{2} \) |
| 71 | \( 1 + (-1.51 + 0.906i)T + (0.471 - 0.881i)T^{2} \) |
| 73 | \( 1 + (0.290 - 0.956i)T^{2} \) |
| 79 | \( 1 + (0.317 - 0.594i)T + (-0.555 - 0.831i)T^{2} \) |
| 83 | \( 1 + (0.998 - 0.0490i)T^{2} \) |
| 89 | \( 1 + (0.995 + 0.0980i)T^{2} \) |
| 97 | \( 1 + (0.382 - 0.923i)T^{2} \) |
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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
−8.475505000989090787523452835868, −8.083932227969937577331471216828, −6.83683698191582498899865369650, −6.16740505689325483665693221689, −5.26052140253291227909851008042, −4.33550170329187956351812308604, −3.57748157197829710207729014039, −3.12102284541281755666609040041, −1.57587737621980811923607115982, −0.831846796716841781983216487102,
1.56805431416053129016135617233, 2.59415175963627749062485799261, 3.95628713451865247749407220528, 4.76254204143102645021806461715, 5.14903420149775742122918492066, 6.14700919025491995076493023404, 6.83642697220030421738116131881, 7.51068798496274728897717068934, 8.269791028660640309737178944322, 8.871847565292255299693771236244