| L(s) = 1 | − 2.53·2-s + 2.11·3-s + 4.40·4-s + 5-s − 5.36·6-s + 2.19·7-s − 6.10·8-s + 1.49·9-s − 2.53·10-s + 11-s + 9.34·12-s + 4.02·13-s − 5.56·14-s + 2.11·15-s + 6.62·16-s + 6.58·17-s − 3.77·18-s − 4.95·19-s + 4.40·20-s + 4.65·21-s − 2.53·22-s − 2.67·23-s − 12.9·24-s + 25-s − 10.1·26-s − 3.19·27-s + 9.69·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 1.22·3-s + 2.20·4-s + 0.447·5-s − 2.19·6-s + 0.830·7-s − 2.15·8-s + 0.497·9-s − 0.800·10-s + 0.301·11-s + 2.69·12-s + 1.11·13-s − 1.48·14-s + 0.547·15-s + 1.65·16-s + 1.59·17-s − 0.889·18-s − 1.13·19-s + 0.986·20-s + 1.01·21-s − 0.539·22-s − 0.557·23-s − 2.63·24-s + 0.200·25-s − 1.99·26-s − 0.615·27-s + 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804282320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804282320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 - 2.11T + 3T^{2} \) |
| 7 | \( 1 - 2.19T + 7T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 19 | \( 1 + 4.95T + 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 - 9.82T + 29T^{2} \) |
| 31 | \( 1 - 6.06T + 31T^{2} \) |
| 37 | \( 1 - 0.228T + 37T^{2} \) |
| 41 | \( 1 + 1.39T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 - 6.69T + 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 79 | \( 1 - 6.89T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 5.76T + 97T^{2} \) |
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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
−8.454547421520684025286221976286, −8.096559733151265623089986127542, −7.48842215192683297723307639326, −6.46191548988733362100797663916, −5.91554832089158781469782905157, −4.54421554252694994018374613918, −3.39692889911649441315538164514, −2.58651384056622940105109989886, −1.72833368371605790806563666107, −1.03731624655869460168071479060,
1.03731624655869460168071479060, 1.72833368371605790806563666107, 2.58651384056622940105109989886, 3.39692889911649441315538164514, 4.54421554252694994018374613918, 5.91554832089158781469782905157, 6.46191548988733362100797663916, 7.48842215192683297723307639326, 8.096559733151265623089986127542, 8.454547421520684025286221976286
