| L(s) = 1 | − 2-s + 3-s + 4-s − 2.47·5-s − 6-s + 3.41·7-s − 8-s + 9-s + 2.47·10-s + 11-s + 12-s + 4.07·13-s − 3.41·14-s − 2.47·15-s + 16-s − 1.29·17-s − 18-s + 4.44·19-s − 2.47·20-s + 3.41·21-s − 22-s + 7.99·23-s − 24-s + 1.12·25-s − 4.07·26-s + 27-s + 3.41·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.10·5-s − 0.408·6-s + 1.29·7-s − 0.353·8-s + 0.333·9-s + 0.782·10-s + 0.301·11-s + 0.288·12-s + 1.13·13-s − 0.912·14-s − 0.639·15-s + 0.250·16-s − 0.313·17-s − 0.235·18-s + 1.01·19-s − 0.553·20-s + 0.744·21-s − 0.213·22-s + 1.66·23-s − 0.204·24-s + 0.225·25-s − 0.799·26-s + 0.192·27-s + 0.645·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.874747140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874747140\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 - 7.99T + 23T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 + 8.46T + 31T^{2} \) |
| 37 | \( 1 + 0.444T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 - 5.92T + 43T^{2} \) |
| 47 | \( 1 - 0.262T + 47T^{2} \) |
| 53 | \( 1 + 8.18T + 53T^{2} \) |
| 59 | \( 1 + 9.27T + 59T^{2} \) |
| 67 | \( 1 - 7.42T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 4.55T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 - 8.45T + 97T^{2} \) |
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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
−8.472827870288492323366524466699, −7.74535305177736541382941928213, −7.39818819498045851890418222421, −6.52949062356894192298281203379, −5.39493731143477915468840000526, −4.55893308462606900953390604317, −3.72164738315245774857934273277, −2.98625083268455318659426363941, −1.71324500774833716198037264899, −0.921289868558479632337230529784,
0.921289868558479632337230529784, 1.71324500774833716198037264899, 2.98625083268455318659426363941, 3.72164738315245774857934273277, 4.55893308462606900953390604317, 5.39493731143477915468840000526, 6.52949062356894192298281203379, 7.39818819498045851890418222421, 7.74535305177736541382941928213, 8.472827870288492323366524466699
