L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.162 − 0.162i)5-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + 0.230i·10-s + (4.08 + 4.08i)11-s + (−2.63 + 2.45i)13-s + 1.00i·14-s − 1.00·16-s + 5.67·17-s + (−0.614 − 0.614i)19-s + (−0.162 − 0.162i)20-s − 5.77·22-s − 3.92·23-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.0728 − 0.0728i)5-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + 0.0728i·10-s + (1.23 + 1.23i)11-s + (−0.732 + 0.681i)13-s + 0.267i·14-s − 0.250·16-s + 1.37·17-s + (−0.141 − 0.141i)19-s + (−0.0364 − 0.0364i)20-s − 1.23·22-s − 0.817·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.350018302\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.350018302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + (2.63 - 2.45i)T \) |
good | 5 | \( 1 + (-0.162 + 0.162i)T - 5iT^{2} \) |
| 11 | \( 1 + (-4.08 - 4.08i)T + 11iT^{2} \) |
| 17 | \( 1 - 5.67T + 17T^{2} \) |
| 19 | \( 1 + (0.614 + 0.614i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.92T + 23T^{2} \) |
| 29 | \( 1 + 6.30iT - 29T^{2} \) |
| 31 | \( 1 + (3.79 + 3.79i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.73 - 1.73i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.60 + 1.60i)T - 41iT^{2} \) |
| 43 | \( 1 - 1.74iT - 43T^{2} \) |
| 47 | \( 1 + (-8.71 - 8.71i)T + 47iT^{2} \) |
| 53 | \( 1 - 9.16iT - 53T^{2} \) |
| 59 | \( 1 + (-4.02 - 4.02i)T + 59iT^{2} \) |
| 61 | \( 1 + 1.44T + 61T^{2} \) |
| 67 | \( 1 + (-1.99 - 1.99i)T + 67iT^{2} \) |
| 71 | \( 1 + (5.82 - 5.82i)T - 71iT^{2} \) |
| 73 | \( 1 + (-0.662 + 0.662i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.96T + 79T^{2} \) |
| 83 | \( 1 + (-7.73 + 7.73i)T - 83iT^{2} \) |
| 89 | \( 1 + (-9.62 - 9.62i)T + 89iT^{2} \) |
| 97 | \( 1 + (-5.78 - 5.78i)T + 97iT^{2} \) |
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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
−9.548042171658868795759881658771, −8.923704219309562640506294311296, −7.59382834456481178033050563419, −7.48996169707993311948661426586, −6.45249966398550392869302808112, −5.63214128256185524586181321727, −4.57180379373419319843383852099, −3.89153804525805378671260747768, −2.23313828650939674533403208230, −1.23760612851350604389682403046,
0.70129334947078685273581560124, 1.92095846429714064631067015953, 3.17618002623370631377204557555, 3.80893878442091919777045932312, 5.15379032470711051907553262091, 5.93121346029357798122895278875, 6.90464549757022875216097570976, 7.84009948222310686925103474382, 8.526033295378960891385827714438, 9.155426651508981307302934847180