| L(s) = 1 | − 2.40·2-s + 0.896·3-s + 3.80·4-s − 0.896·5-s − 2.15·6-s − 4.34·8-s − 2.19·9-s + 2.15·10-s + 1.30·11-s + 3.40·12-s − 0.896·13-s − 0.803·15-s + 2.85·16-s + 2.65·17-s + 5.29·18-s − 3.00·19-s − 3.40·20-s − 3.14·22-s + 5.35·23-s − 3.89·24-s − 4.19·25-s + 2.15·26-s − 4.65·27-s + 0.155·29-s + 1.93·30-s + 8.51·31-s + 1.80·32-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 0.517·3-s + 1.90·4-s − 0.400·5-s − 0.881·6-s − 1.53·8-s − 0.732·9-s + 0.682·10-s + 0.393·11-s + 0.984·12-s − 0.248·13-s − 0.207·15-s + 0.713·16-s + 0.644·17-s + 1.24·18-s − 0.690·19-s − 0.762·20-s − 0.670·22-s + 1.11·23-s − 0.794·24-s − 0.839·25-s + 0.423·26-s − 0.896·27-s + 0.0288·29-s + 0.353·30-s + 1.52·31-s + 0.319·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 - 0.896T + 3T^{2} \) |
| 5 | \( 1 + 0.896T + 5T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 + 0.896T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 + 3.00T + 19T^{2} \) |
| 23 | \( 1 - 5.35T + 23T^{2} \) |
| 29 | \( 1 - 0.155T + 29T^{2} \) |
| 31 | \( 1 - 8.51T + 31T^{2} \) |
| 37 | \( 1 + 2.06T + 37T^{2} \) |
| 43 | \( 1 + 4.43T + 43T^{2} \) |
| 47 | \( 1 + 2.25T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 - 2.66T + 61T^{2} \) |
| 67 | \( 1 - 4.21T + 67T^{2} \) |
| 71 | \( 1 - 1.45T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.664283888643319701423259317722, −8.219023620706692056362307429024, −7.56545469571428004097199765981, −6.75146924391849716507178295601, −5.93076208427744534940511858207, −4.63567764919776045539140174906, −3.35164832487139713786886452159, −2.51972438404041901459294596558, −1.36529399632823505451369982604, 0,
1.36529399632823505451369982604, 2.51972438404041901459294596558, 3.35164832487139713786886452159, 4.63567764919776045539140174906, 5.93076208427744534940511858207, 6.75146924391849716507178295601, 7.56545469571428004097199765981, 8.219023620706692056362307429024, 8.664283888643319701423259317722
