| L(s) = 1 | + 2-s + 0.0851·3-s + 4-s − 0.378·5-s + 0.0851·6-s + 4.38·7-s + 8-s − 2.99·9-s − 0.378·10-s + 0.576·11-s + 0.0851·12-s + 6.42·13-s + 4.38·14-s − 0.0322·15-s + 16-s + 2.26·17-s − 2.99·18-s + 0.566·19-s − 0.378·20-s + 0.373·21-s + 0.576·22-s − 23-s + 0.0851·24-s − 4.85·25-s + 6.42·26-s − 0.510·27-s + 4.38·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0491·3-s + 0.5·4-s − 0.169·5-s + 0.0347·6-s + 1.65·7-s + 0.353·8-s − 0.997·9-s − 0.119·10-s + 0.173·11-s + 0.0245·12-s + 1.78·13-s + 1.17·14-s − 0.00832·15-s + 0.250·16-s + 0.548·17-s − 0.705·18-s + 0.129·19-s − 0.0846·20-s + 0.0815·21-s + 0.122·22-s − 0.208·23-s + 0.0173·24-s − 0.971·25-s + 1.25·26-s − 0.0981·27-s + 0.829·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.224791369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.224791369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0851T + 3T^{2} \) |
| 5 | \( 1 + 0.378T + 5T^{2} \) |
| 7 | \( 1 - 4.38T + 7T^{2} \) |
| 11 | \( 1 - 0.576T + 11T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 - 0.566T + 19T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 + 0.734T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 - 5.59T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 9.86T + 59T^{2} \) |
| 61 | \( 1 - 7.35T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 0.336T + 71T^{2} \) |
| 73 | \( 1 + 8.53T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 1.38T + 83T^{2} \) |
| 89 | \( 1 - 4.53T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
| 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
−7.925064504303886929300360077671, −7.63526574877615449316316544235, −6.32581040111096315271560937942, −5.92099243532128995873825524831, −5.20406817234026510949591120213, −4.42516461988523459814480863772, −3.75370902066259960816531383473, −2.91555892051282472847166107262, −1.86807503434964719128039678553, −1.05492997055256161211859724977,
1.05492997055256161211859724977, 1.86807503434964719128039678553, 2.91555892051282472847166107262, 3.75370902066259960816531383473, 4.42516461988523459814480863772, 5.20406817234026510949591120213, 5.92099243532128995873825524831, 6.32581040111096315271560937942, 7.63526574877615449316316544235, 7.925064504303886929300360077671
