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 + (0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + (−0.499 − 0.866i)12-s + (0.499 − 0.866i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (−0.499 − 0.866i)20-s − 0.999·21-s + ⋯ |
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 + (0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + (−0.499 − 0.866i)12-s + (0.499 − 0.866i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (−0.499 − 0.866i)20-s − 0.999·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4308082118\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4308082118\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−12.42240255709205703010474074280, −11.81595518153823601415363561867, −10.91868286944090228996622822922, −10.36180860361938804868043823832, −9.144102951616797963774964463606, −8.310140351924579553413081198973, −6.85589382162551048678061698581, −5.24204552749825469170490668341, −3.96139403068010698317325041094, −2.69175475848050787785423695874,
1.22605061796298853307315365905, 4.43443877732487047943431891381, 5.44771769571650859231311693028, 6.73967792274990793223112557608, 7.72987037568855243508650243723, 8.286011076617848398893161331147, 9.612274076556058093848870083957, 10.90274568229085661893260839840, 11.73600259250111435577981040744, 13.04264661200903517342027650101