| L(s) = 1 | + 1.14·2-s − 0.692·4-s + 3.48·5-s − 3.35·7-s − 3.07·8-s + 3.98·10-s − 2.97·11-s − 3.83·14-s − 2.13·16-s − 6.91·17-s + 0.295·19-s − 2.41·20-s − 3.40·22-s − 2.46·23-s + 7.15·25-s + 2.32·28-s − 0.609·29-s − 9.93·31-s + 3.71·32-s − 7.91·34-s − 11.7·35-s + 8.50·37-s + 0.338·38-s − 10.7·40-s − 7.30·41-s − 6.60·43-s + 2.06·44-s + ⋯ |
| L(s) = 1 | + 0.808·2-s − 0.346·4-s + 1.55·5-s − 1.26·7-s − 1.08·8-s + 1.26·10-s − 0.897·11-s − 1.02·14-s − 0.534·16-s − 1.67·17-s + 0.0678·19-s − 0.539·20-s − 0.726·22-s − 0.514·23-s + 1.43·25-s + 0.439·28-s − 0.113·29-s − 1.78·31-s + 0.656·32-s − 1.35·34-s − 1.97·35-s + 1.39·37-s + 0.0548·38-s − 1.69·40-s − 1.14·41-s − 1.00·43-s + 0.310·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.14T + 2T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 + 2.97T + 11T^{2} \) |
| 17 | \( 1 + 6.91T + 17T^{2} \) |
| 19 | \( 1 - 0.295T + 19T^{2} \) |
| 23 | \( 1 + 2.46T + 23T^{2} \) |
| 29 | \( 1 + 0.609T + 29T^{2} \) |
| 31 | \( 1 + 9.93T + 31T^{2} \) |
| 37 | \( 1 - 8.50T + 37T^{2} \) |
| 41 | \( 1 + 7.30T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 - 4.63T + 47T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 + 0.816T + 59T^{2} \) |
| 61 | \( 1 - 5.08T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 8.74T + 73T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 7.77T + 89T^{2} \) |
| 97 | \( 1 + 2.56T + 97T^{2} \) |
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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
−9.197738093181764394814642433431, −8.575248920038490465527608670506, −7.09937030856506512836119028291, −6.30455948181440194303592987238, −5.75399653231567427581867711953, −5.04039806920510030508921723199, −3.97963125442985336935949456286, −2.91384964417990387619787980262, −2.12687476191771707287867729841, 0,
2.12687476191771707287867729841, 2.91384964417990387619787980262, 3.97963125442985336935949456286, 5.04039806920510030508921723199, 5.75399653231567427581867711953, 6.30455948181440194303592987238, 7.09937030856506512836119028291, 8.575248920038490465527608670506, 9.197738093181764394814642433431
