| L(s) = 1 | + 2-s − 3-s + 4-s + 1.78·5-s − 6-s + 0.126·7-s + 8-s + 9-s + 1.78·10-s − 2.47·11-s − 12-s + 4.19·13-s + 0.126·14-s − 1.78·15-s + 16-s + 7.66·17-s + 18-s + 0.126·19-s + 1.78·20-s − 0.126·21-s − 2.47·22-s + 23-s − 24-s − 1.79·25-s + 4.19·26-s − 27-s + 0.126·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.800·5-s − 0.408·6-s + 0.0478·7-s + 0.353·8-s + 0.333·9-s + 0.565·10-s − 0.745·11-s − 0.288·12-s + 1.16·13-s + 0.0338·14-s − 0.461·15-s + 0.250·16-s + 1.85·17-s + 0.235·18-s + 0.0290·19-s + 0.400·20-s − 0.0276·21-s − 0.527·22-s + 0.208·23-s − 0.204·24-s − 0.359·25-s + 0.823·26-s − 0.192·27-s + 0.0239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.245676925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.245676925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 1.78T + 5T^{2} \) |
| 7 | \( 1 - 0.126T + 7T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 - 7.66T + 17T^{2} \) |
| 19 | \( 1 - 0.126T + 19T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 - 0.736T + 37T^{2} \) |
| 41 | \( 1 + 2.59T + 41T^{2} \) |
| 43 | \( 1 + 6.10T + 43T^{2} \) |
| 47 | \( 1 + 6.46T + 47T^{2} \) |
| 53 | \( 1 - 6.06T + 53T^{2} \) |
| 59 | \( 1 + 3.99T + 59T^{2} \) |
| 61 | \( 1 - 6.03T + 61T^{2} \) |
| 67 | \( 1 - 8.50T + 67T^{2} \) |
| 71 | \( 1 - 0.604T + 71T^{2} \) |
| 73 | \( 1 - 3.57T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 5.53T + 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 9.09T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.188595454983906867891333338951, −7.74488530850097262552236539979, −6.59722175655101401151623082079, −6.17841452944349235645423143508, −5.33905891410561115365331215249, −5.05195477351417119930915873180, −3.79752454289482616995357452928, −3.12477796909336906577067401104, −1.96996952547322643964347186666, −1.01806165169176687269283320691,
1.01806165169176687269283320691, 1.96996952547322643964347186666, 3.12477796909336906577067401104, 3.79752454289482616995357452928, 5.05195477351417119930915873180, 5.33905891410561115365331215249, 6.17841452944349235645423143508, 6.59722175655101401151623082079, 7.74488530850097262552236539979, 8.188595454983906867891333338951
