| L(s) = 1 | + 0.554·2-s + 3-s − 1.69·4-s + 5-s + 0.554·6-s − 4.49·7-s − 2.04·8-s + 9-s + 0.554·10-s − 2·11-s − 1.69·12-s − 2.49·14-s + 15-s + 2.24·16-s + 2.91·17-s + 0.554·18-s + 2.04·19-s − 1.69·20-s − 4.49·21-s − 1.10·22-s + 4.35·23-s − 2.04·24-s + 25-s + 27-s + 7.60·28-s − 0.713·29-s + 0.554·30-s + ⋯ |
| L(s) = 1 | + 0.392·2-s + 0.577·3-s − 0.846·4-s + 0.447·5-s + 0.226·6-s − 1.69·7-s − 0.724·8-s + 0.333·9-s + 0.175·10-s − 0.603·11-s − 0.488·12-s − 0.666·14-s + 0.258·15-s + 0.561·16-s + 0.706·17-s + 0.130·18-s + 0.470·19-s − 0.378·20-s − 0.980·21-s − 0.236·22-s + 0.908·23-s − 0.418·24-s + 0.200·25-s + 0.192·27-s + 1.43·28-s − 0.132·29-s + 0.101·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.717911327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717911327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.554T + 2T^{2} \) |
| 7 | \( 1 + 4.49T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 - 4.35T + 23T^{2} \) |
| 29 | \( 1 + 0.713T + 29T^{2} \) |
| 31 | \( 1 + 5.89T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 4.21T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 8.71T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 1.36T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 - 1.78T + 89T^{2} \) |
| 97 | \( 1 - 1.20T + 97T^{2} \) |
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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
−9.078422156874565998922309401724, −8.256003247096767651127851633795, −7.36462705059633305737257337880, −6.50587789468272746302218149466, −5.69927715829499031776921738680, −5.04457620295652222409079037049, −3.86337885775416970603872323668, −3.26855201252120347298149420035, −2.52553185049546488426323602122, −0.75151954430084235836452449150,
0.75151954430084235836452449150, 2.52553185049546488426323602122, 3.26855201252120347298149420035, 3.86337885775416970603872323668, 5.04457620295652222409079037049, 5.69927715829499031776921738680, 6.50587789468272746302218149466, 7.36462705059633305737257337880, 8.256003247096767651127851633795, 9.078422156874565998922309401724
