| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 1.23·7-s + 8-s + 9-s + 10-s − 11-s + 12-s + 4.87·13-s + 1.23·14-s + 15-s + 16-s − 17-s + 18-s − 0.872·19-s + 20-s + 1.23·21-s − 22-s + 9.36·23-s + 24-s + 25-s + 4.87·26-s + 27-s + 1.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.465·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.35·13-s + 0.328·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.200·19-s + 0.223·20-s + 0.268·21-s − 0.213·22-s + 1.95·23-s + 0.204·24-s + 0.200·25-s + 0.955·26-s + 0.192·27-s + 0.232·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.132058656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.132058656\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 1.23T + 7T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 19 | \( 1 + 0.872T + 19T^{2} \) |
| 23 | \( 1 - 9.36T + 23T^{2} \) |
| 29 | \( 1 - 3.64T + 29T^{2} \) |
| 31 | \( 1 - 2.10T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 8.59T + 41T^{2} \) |
| 43 | \( 1 + 0.979T + 43T^{2} \) |
| 47 | \( 1 - 8.59T + 47T^{2} \) |
| 53 | \( 1 + 7.92T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 - 0.00337T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 8.62T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 - 1.28T + 89T^{2} \) |
| 97 | \( 1 - 7.85T + 97T^{2} \) |
| 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
−8.321910112857824932456995697194, −7.24727841989675393426432259146, −6.72204802523987022467235032347, −5.95550323328111792265139533768, −5.10098102751485688243150120926, −4.58806417929120672301077205071, −3.52828307275221589365311999329, −3.00477171615577425608962420301, −1.97223578720233713731024668169, −1.16684938274224499266072993892,
1.16684938274224499266072993892, 1.97223578720233713731024668169, 3.00477171615577425608962420301, 3.52828307275221589365311999329, 4.58806417929120672301077205071, 5.10098102751485688243150120926, 5.95550323328111792265139533768, 6.72204802523987022467235032347, 7.24727841989675393426432259146, 8.321910112857824932456995697194
