| L(s) = 1 | − 2-s − 0.679·3-s + 4-s − 2.19·5-s + 0.679·6-s + 7-s − 8-s − 2.53·9-s + 2.19·10-s + 2.71·11-s − 0.679·12-s − 4.48·13-s − 14-s + 1.48·15-s + 16-s − 5.92·17-s + 2.53·18-s − 5.99·19-s − 2.19·20-s − 0.679·21-s − 2.71·22-s − 7.05·23-s + 0.679·24-s − 0.201·25-s + 4.48·26-s + 3.76·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.392·3-s + 0.5·4-s − 0.979·5-s + 0.277·6-s + 0.377·7-s − 0.353·8-s − 0.846·9-s + 0.692·10-s + 0.818·11-s − 0.196·12-s − 1.24·13-s − 0.267·14-s + 0.384·15-s + 0.250·16-s − 1.43·17-s + 0.598·18-s − 1.37·19-s − 0.489·20-s − 0.148·21-s − 0.578·22-s − 1.47·23-s + 0.138·24-s − 0.0402·25-s + 0.879·26-s + 0.724·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06518867915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06518867915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 0.679T + 3T^{2} \) |
| 5 | \( 1 + 2.19T + 5T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 + 5.92T + 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 23 | \( 1 + 7.05T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 9.81T + 31T^{2} \) |
| 37 | \( 1 + 9.60T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 - 2.75T + 53T^{2} \) |
| 59 | \( 1 + 6.34T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 5.96T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 5.57T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
| 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
−8.084868610331754063389332616772, −7.41238277268350174377980036815, −6.87123415631197567083661802792, −6.02257960022808441346712597109, −5.34006138501229241926319855447, −4.19538217348632671458749557620, −3.89725818094852973720208576091, −2.47740375586025092270956381091, −1.87518918834090817482796655370, −0.14217994957294939144430581589,
0.14217994957294939144430581589, 1.87518918834090817482796655370, 2.47740375586025092270956381091, 3.89725818094852973720208576091, 4.19538217348632671458749557620, 5.34006138501229241926319855447, 6.02257960022808441346712597109, 6.87123415631197567083661802792, 7.41238277268350174377980036815, 8.084868610331754063389332616772
