| L(s) = 1 | − 2.77·2-s − 0.377·3-s + 5.70·4-s − 1.07·5-s + 1.04·6-s − 3.82·7-s − 10.2·8-s − 2.85·9-s + 2.99·10-s + 11-s − 2.15·12-s + 6.34·13-s + 10.6·14-s + 0.407·15-s + 17.1·16-s + 17-s + 7.93·18-s + 2.71·19-s − 6.16·20-s + 1.44·21-s − 2.77·22-s + 3.31·23-s + 3.88·24-s − 3.83·25-s − 17.6·26-s + 2.21·27-s − 21.8·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 0.217·3-s + 2.85·4-s − 0.482·5-s + 0.427·6-s − 1.44·7-s − 3.64·8-s − 0.952·9-s + 0.947·10-s + 0.301·11-s − 0.621·12-s + 1.75·13-s + 2.83·14-s + 0.105·15-s + 4.29·16-s + 0.242·17-s + 1.87·18-s + 0.622·19-s − 1.37·20-s + 0.314·21-s − 0.591·22-s + 0.691·23-s + 0.793·24-s − 0.767·25-s − 3.45·26-s + 0.425·27-s − 4.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4860198612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4860198612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 + 0.377T + 3T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 19 | \( 1 - 2.71T + 19T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 + 0.300T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 - 6.74T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 9.86T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 0.972T + 73T^{2} \) |
| 79 | \( 1 + 8.47T + 79T^{2} \) |
| 83 | \( 1 + 8.08T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 5.49T + 97T^{2} \) |
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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
−8.208596746349812438404563342914, −7.10411472084301498907804024659, −6.77185804269465948231876115724, −5.98122937732782028408609207471, −5.65901092793526329408536123630, −3.82462025803753228075687178379, −3.20343059990002911014891626080, −2.59396264861161329261869487856, −1.23734420280587126249513213589, −0.53153361161524971674468333096,
0.53153361161524971674468333096, 1.23734420280587126249513213589, 2.59396264861161329261869487856, 3.20343059990002911014891626080, 3.82462025803753228075687178379, 5.65901092793526329408536123630, 5.98122937732782028408609207471, 6.77185804269465948231876115724, 7.10411472084301498907804024659, 8.208596746349812438404563342914
