| L(s) = 1 | − 2-s − 3-s + 4-s + 2.03·5-s + 6-s + 2.20·7-s − 8-s + 9-s − 2.03·10-s + 3.73·11-s − 12-s − 3.23·13-s − 2.20·14-s − 2.03·15-s + 16-s − 2.81·17-s − 18-s + 3.23·19-s + 2.03·20-s − 2.20·21-s − 3.73·22-s − 6·23-s + 24-s − 0.854·25-s + 3.23·26-s − 27-s + 2.20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.910·5-s + 0.408·6-s + 0.834·7-s − 0.353·8-s + 0.333·9-s − 0.643·10-s + 1.12·11-s − 0.288·12-s − 0.897·13-s − 0.590·14-s − 0.525·15-s + 0.250·16-s − 0.683·17-s − 0.235·18-s + 0.742·19-s + 0.455·20-s − 0.482·21-s − 0.797·22-s − 1.25·23-s + 0.204·24-s − 0.170·25-s + 0.634·26-s − 0.192·27-s + 0.417·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.546278981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.546278981\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 - 2.03T + 5T^{2} \) |
| 7 | \( 1 - 2.20T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 37 | \( 1 - 9.56T + 37T^{2} \) |
| 41 | \( 1 - 4.84T + 41T^{2} \) |
| 43 | \( 1 + 4.21T + 43T^{2} \) |
| 47 | \( 1 - 5.44T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 4.79T + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 9.45T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 4.67T + 97T^{2} \) |
| 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
−7.958323163987396814509238232154, −7.57725416443098873275184052754, −6.53073574059064235848944339184, −6.20192373151946964163057537415, −5.32588994785760937885349188431, −4.62870136818759010548435717067, −3.71716038854975201067991670818, −2.31564181477066832563656501084, −1.80717145070546964653971624203, −0.77347619178617593049162002645,
0.77347619178617593049162002645, 1.80717145070546964653971624203, 2.31564181477066832563656501084, 3.71716038854975201067991670818, 4.62870136818759010548435717067, 5.32588994785760937885349188431, 6.20192373151946964163057537415, 6.53073574059064235848944339184, 7.57725416443098873275184052754, 7.958323163987396814509238232154
