| L(s) = 1 | − 0.101·3-s + 2.19·5-s − 7-s − 2.98·9-s + 0.674·11-s − 5.30·13-s − 0.222·15-s + 3.87·17-s + 0.995·19-s + 0.101·21-s + 3.36·23-s − 0.175·25-s + 0.607·27-s + 0.134·29-s + 3.11·31-s − 0.0683·33-s − 2.19·35-s + 2.07·37-s + 0.537·39-s − 3.76·41-s + 3.61·43-s − 6.56·45-s + 6.68·47-s + 49-s − 0.392·51-s + 2.93·53-s + 1.48·55-s + ⋯ |
| L(s) = 1 | − 0.0585·3-s + 0.982·5-s − 0.377·7-s − 0.996·9-s + 0.203·11-s − 1.47·13-s − 0.0574·15-s + 0.939·17-s + 0.228·19-s + 0.0221·21-s + 0.701·23-s − 0.0351·25-s + 0.116·27-s + 0.0249·29-s + 0.559·31-s − 0.0118·33-s − 0.371·35-s + 0.341·37-s + 0.0860·39-s − 0.588·41-s + 0.550·43-s − 0.978·45-s + 0.975·47-s + 0.142·49-s − 0.0550·51-s + 0.403·53-s + 0.199·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.827980151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827980151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 0.101T + 3T^{2} \) |
| 5 | \( 1 - 2.19T + 5T^{2} \) |
| 11 | \( 1 - 0.674T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 - 0.995T + 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 - 0.134T + 29T^{2} \) |
| 31 | \( 1 - 3.11T + 31T^{2} \) |
| 37 | \( 1 - 2.07T + 37T^{2} \) |
| 41 | \( 1 + 3.76T + 41T^{2} \) |
| 43 | \( 1 - 3.61T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 - 2.93T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 4.30T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 0.856T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
| 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
−8.613370970782822113870712620463, −7.81948045354784822486609530403, −6.99194468327022468428404480182, −6.28079237062294120031761573612, −5.43323934996912889299405306937, −5.10871391567601364423613271715, −3.80808735734032998580343181950, −2.80087036112803417622322008558, −2.22089452836392355983039616843, −0.78086879841641222774961584513,
0.78086879841641222774961584513, 2.22089452836392355983039616843, 2.80087036112803417622322008558, 3.80808735734032998580343181950, 5.10871391567601364423613271715, 5.43323934996912889299405306937, 6.28079237062294120031761573612, 6.99194468327022468428404480182, 7.81948045354784822486609530403, 8.613370970782822113870712620463
