| L(s) = 1 | + 3-s − 3.82·7-s + 9-s + 5.82i·11-s + 4.82i·13-s + (2.82 − 3i)17-s − 3·19-s − 3.82·21-s + 8.48·23-s + 27-s − 4.17i·29-s − 6.82i·31-s + 5.82i·33-s − 9.48·37-s + 4.82i·39-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.44·7-s + 0.333·9-s + 1.75i·11-s + 1.33i·13-s + (0.685 − 0.727i)17-s − 0.688·19-s − 0.835·21-s + 1.76·23-s + 0.192·27-s − 0.774i·29-s − 1.22i·31-s + 1.01i·33-s − 1.55·37-s + 0.773i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8153951274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8153951274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (-2.82 + 3i)T \) |
| good | 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 - 5.82iT - 11T^{2} \) |
| 13 | \( 1 - 4.82iT - 13T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 4.17iT - 29T^{2} \) |
| 31 | \( 1 + 6.82iT - 31T^{2} \) |
| 37 | \( 1 + 9.48T + 37T^{2} \) |
| 41 | \( 1 + 5.82iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 8.65iT - 47T^{2} \) |
| 53 | \( 1 - 8.31iT - 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 2.48iT - 67T^{2} \) |
| 71 | \( 1 + 9.65iT - 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.782987784814484791421379213753, −7.59332474002942884915554806609, −7.10847216298586905375817959848, −6.65411268516153300949792556297, −5.75786664790723755016247601497, −4.57761541081768149005499085206, −4.21047214154996545839787306986, −3.11137401045793822291636689445, −2.48505724768108668293893013143, −1.47225072811284237007621459156,
0.20115594052814208881589331821, 1.30541797554641504502609500409, 2.89347928985592288467747295399, 3.24783050448172473131132408252, 3.70811950182923220650863276691, 5.21295498222944838269370899296, 5.65831078316050593237943889008, 6.61845776836339132768493506396, 7.01303579115883140564212625375, 8.166363640830165973601574933906
