| L(s) = 1 | − 0.973·2-s + 1.32·3-s − 1.05·4-s − 5-s − 1.28·6-s + 3.84·7-s + 2.97·8-s − 1.24·9-s + 0.973·10-s + 5.53·11-s − 1.39·12-s + 3.19·13-s − 3.74·14-s − 1.32·15-s − 0.790·16-s + 3.05·17-s + 1.21·18-s + 1.27·19-s + 1.05·20-s + 5.09·21-s − 5.39·22-s + 4.69·23-s + 3.93·24-s + 25-s − 3.10·26-s − 5.62·27-s − 4.04·28-s + ⋯ |
| L(s) = 1 | − 0.688·2-s + 0.763·3-s − 0.525·4-s − 0.447·5-s − 0.526·6-s + 1.45·7-s + 1.05·8-s − 0.416·9-s + 0.307·10-s + 1.66·11-s − 0.401·12-s + 0.885·13-s − 1.00·14-s − 0.341·15-s − 0.197·16-s + 0.741·17-s + 0.286·18-s + 0.292·19-s + 0.235·20-s + 1.11·21-s − 1.14·22-s + 0.979·23-s + 0.802·24-s + 0.200·25-s − 0.609·26-s − 1.08·27-s − 0.764·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.719239484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719239484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.973T + 2T^{2} \) |
| 3 | \( 1 - 1.32T + 3T^{2} \) |
| 7 | \( 1 - 3.84T + 7T^{2} \) |
| 11 | \( 1 - 5.53T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 4.69T + 23T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 + 8.91T + 31T^{2} \) |
| 37 | \( 1 + 1.99T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 8.20T + 47T^{2} \) |
| 53 | \( 1 - 2.49T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 + 5.61T + 61T^{2} \) |
| 67 | \( 1 + 2.67T + 67T^{2} \) |
| 71 | \( 1 + 4.59T + 71T^{2} \) |
| 73 | \( 1 - 3.17T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 7.28T + 97T^{2} \) |
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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
−8.961879493281653614708733753500, −8.661261987866155245742786147407, −7.60101711014168649058555688293, −7.46858906768931991949497159942, −5.92229345759316124698588929074, −5.04591328389360548171384648243, −3.99144005305464834196563957556, −3.53343222367073080372041378119, −1.84904972884237450783268015635, −1.05281426382864700345048219551,
1.05281426382864700345048219551, 1.84904972884237450783268015635, 3.53343222367073080372041378119, 3.99144005305464834196563957556, 5.04591328389360548171384648243, 5.92229345759316124698588929074, 7.46858906768931991949497159942, 7.60101711014168649058555688293, 8.661261987866155245742786147407, 8.961879493281653614708733753500
