| L(s) = 1 | − 0.920·2-s + 0.898·3-s − 1.15·4-s − 2.76·5-s − 0.827·6-s + 3.49·7-s + 2.90·8-s − 2.19·9-s + 2.54·10-s − 11-s − 1.03·12-s − 3.21·14-s − 2.48·15-s − 0.362·16-s − 2.81·17-s + 2.01·18-s + 0.317·19-s + 3.18·20-s + 3.13·21-s + 0.920·22-s − 1.29·23-s + 2.60·24-s + 2.63·25-s − 4.66·27-s − 4.02·28-s + 2.92·29-s + 2.28·30-s + ⋯ |
| L(s) = 1 | − 0.650·2-s + 0.518·3-s − 0.576·4-s − 1.23·5-s − 0.337·6-s + 1.32·7-s + 1.02·8-s − 0.730·9-s + 0.804·10-s − 0.301·11-s − 0.299·12-s − 0.858·14-s − 0.641·15-s − 0.0906·16-s − 0.681·17-s + 0.475·18-s + 0.0729·19-s + 0.712·20-s + 0.685·21-s + 0.196·22-s − 0.270·23-s + 0.532·24-s + 0.527·25-s − 0.898·27-s − 0.761·28-s + 0.542·29-s + 0.417·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8900907215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8900907215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.920T + 2T^{2} \) |
| 3 | \( 1 - 0.898T + 3T^{2} \) |
| 5 | \( 1 + 2.76T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 - 0.317T + 19T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 - 5.06T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 4.96T + 41T^{2} \) |
| 43 | \( 1 + 0.952T + 43T^{2} \) |
| 47 | \( 1 + 5.36T + 47T^{2} \) |
| 53 | \( 1 + 6.51T + 53T^{2} \) |
| 59 | \( 1 - 7.99T + 59T^{2} \) |
| 61 | \( 1 + 7.39T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.934099617906124957429763873443, −8.373905990636632359779664739657, −7.904261920183095113562076460780, −7.48249935608386071638367559554, −6.06535787471462380887688731042, −4.77787890008018469466647265848, −4.45490843267925871388580156766, −3.36675772466124504704050909374, −2.13522853464325324550002944841, −0.68964468036382792136680583733,
0.68964468036382792136680583733, 2.13522853464325324550002944841, 3.36675772466124504704050909374, 4.45490843267925871388580156766, 4.77787890008018469466647265848, 6.06535787471462380887688731042, 7.48249935608386071638367559554, 7.904261920183095113562076460780, 8.373905990636632359779664739657, 8.934099617906124957429763873443
