| L(s) = 1 | − 0.726i·2-s + 1.38·3-s + 3.47·4-s + 4·5-s − 1.00i·6-s + 9.95i·7-s − 5.42i·8-s − 7.09·9-s − 2.90i·10-s + 4.79·12-s − 5.25i·13-s + 7.23·14-s + 5.52·15-s + 9.94·16-s − 15.2i·17-s + 5.15i·18-s + ⋯ |
| L(s) = 1 | − 0.363i·2-s + 0.460·3-s + 0.868·4-s + 0.800·5-s − 0.167i·6-s + 1.42i·7-s − 0.678i·8-s − 0.787·9-s − 0.290i·10-s + 0.399·12-s − 0.404i·13-s + 0.516·14-s + 0.368·15-s + 0.621·16-s − 0.898i·17-s + 0.286i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.97412 - 0.218885i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97412 - 0.218885i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 0.726iT - 4T^{2} \) |
| 3 | \( 1 - 1.38T + 9T^{2} \) |
| 5 | \( 1 - 4T + 25T^{2} \) |
| 7 | \( 1 - 9.95iT - 49T^{2} \) |
| 13 | \( 1 + 5.25iT - 169T^{2} \) |
| 17 | \( 1 + 15.2iT - 289T^{2} \) |
| 19 | \( 1 + 2.07iT - 361T^{2} \) |
| 23 | \( 1 + 2.76T + 529T^{2} \) |
| 29 | \( 1 + 28.4iT - 841T^{2} \) |
| 31 | \( 1 + 7.12T + 961T^{2} \) |
| 37 | \( 1 + 40.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 70.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 27.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 11.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 2.27T + 3.48e3T^{2} \) |
| 61 | \( 1 + 22.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 38.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 76.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 102. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 3.93iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 83.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 123.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 77.4T + 9.40e3T^{2} \) |
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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
−13.10780184907596737458130782280, −11.99270694416533961397466196881, −11.32705173013387399728224251912, −9.943433872675401186854652682143, −9.086041492745447109027842818790, −7.900550399211410282956992402349, −6.30487879563592364325666468134, −5.46066216430236000831117008806, −3.00605157048239202890955480875, −2.16115369034762161717548670099,
1.92332128323032736125812850306, 3.61988003535187609757778686332, 5.56706524303702039138541083858, 6.69825847467274651509432689244, 7.68334536665281264052958723841, 8.879142777423662526629275551819, 10.29842618935403553320467484534, 10.93983234661696636053166761897, 12.26245876059873233647934925115, 13.68678868615251856083302143565
