| L(s) = 1 | + 0.174·2-s − 1.66·3-s − 1.96·4-s + 1.49·5-s − 0.290·6-s − 1.90·7-s − 0.694·8-s − 0.242·9-s + 0.260·10-s − 11-s + 3.27·12-s + 4.35·13-s − 0.332·14-s − 2.47·15-s + 3.81·16-s + 17-s − 0.0424·18-s − 6.83·19-s − 2.93·20-s + 3.16·21-s − 0.174·22-s + 6.95·23-s + 1.15·24-s − 2.77·25-s + 0.762·26-s + 5.38·27-s + 3.74·28-s + ⋯ |
| L(s) = 1 | + 0.123·2-s − 0.958·3-s − 0.984·4-s + 0.667·5-s − 0.118·6-s − 0.719·7-s − 0.245·8-s − 0.0809·9-s + 0.0825·10-s − 0.301·11-s + 0.944·12-s + 1.20·13-s − 0.0889·14-s − 0.639·15-s + 0.954·16-s + 0.242·17-s − 0.0100·18-s − 1.56·19-s − 0.656·20-s + 0.689·21-s − 0.0372·22-s + 1.44·23-s + 0.235·24-s − 0.554·25-s + 0.149·26-s + 1.03·27-s + 0.708·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8230312417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8230312417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 0.174T + 2T^{2} \) |
| 3 | \( 1 + 1.66T + 3T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 13 | \( 1 - 4.35T + 13T^{2} \) |
| 19 | \( 1 + 6.83T + 19T^{2} \) |
| 23 | \( 1 - 6.95T + 23T^{2} \) |
| 29 | \( 1 - 9.83T + 29T^{2} \) |
| 31 | \( 1 - 7.01T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 47 | \( 1 - 0.318T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 7.43T + 59T^{2} \) |
| 61 | \( 1 + 4.53T + 61T^{2} \) |
| 67 | \( 1 + 8.51T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 - 3.71T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.042474483552855332460640868548, −6.68983560264370696606649764855, −6.30151558130655777716001437173, −5.88861470018951505816867123289, −4.91581592129138891701309559260, −4.65982097306887514634950127732, −3.47329874781490255495842644171, −2.90356413091742242970716587510, −1.49717967151057340488304641459, −0.48826443934715437600051422011,
0.48826443934715437600051422011, 1.49717967151057340488304641459, 2.90356413091742242970716587510, 3.47329874781490255495842644171, 4.65982097306887514634950127732, 4.91581592129138891701309559260, 5.88861470018951505816867123289, 6.30151558130655777716001437173, 6.68983560264370696606649764855, 8.042474483552855332460640868548
