| L(s) = 1 | − 1.41i·2-s − 0.317i·3-s − 2.00·4-s − 5-s − 0.449·6-s − 2.89·7-s + 2.82i·8-s + 8.89·9-s + 1.41i·10-s − 5.10·11-s + 0.635i·12-s − 0.174i·13-s + 4.09i·14-s + 0.317i·15-s + 4.00·16-s − 11.8·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.105i·3-s − 0.500·4-s − 0.200·5-s − 0.0749·6-s − 0.414·7-s + 0.353i·8-s + 0.988·9-s + 0.141i·10-s − 0.463·11-s + 0.0529i·12-s − 0.0134i·13-s + 0.292i·14-s + 0.0211i·15-s + 0.250·16-s − 0.699·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7500191370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7500191370\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.317iT - 9T^{2} \) |
| 5 | \( 1 + T + 25T^{2} \) |
| 7 | \( 1 + 2.89T + 49T^{2} \) |
| 11 | \( 1 + 5.10T + 121T^{2} \) |
| 13 | \( 1 + 0.174iT - 169T^{2} \) |
| 17 | \( 1 + 11.8T + 289T^{2} \) |
| 23 | \( 1 + 17.0T + 529T^{2} \) |
| 29 | \( 1 - 44.5iT - 841T^{2} \) |
| 31 | \( 1 - 31.1iT - 961T^{2} \) |
| 37 | \( 1 + 21.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 54.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 37.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 81.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 55.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 34.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 76.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 118. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 75.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 29.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 66.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 30.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 10.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 148. iT - 9.40e3T^{2} \) |
| 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
−10.44895886163271516646243801829, −9.633295195862110667461960531350, −8.843196208804205606022009741539, −7.78051196449810483673830162411, −6.97064145008781681089415588150, −5.84290674813267589589922719877, −4.65361190657621217036803670537, −3.82325923405605806575322411178, −2.63547901292122259358084912359, −1.37343174383498841069561776729,
0.27107160892860222670693756454, 2.17957430998056915006776979044, 3.81238123437747049055135582912, 4.48892659030209703971524810850, 5.72048269293709133117201834305, 6.51609742379758545177083179368, 7.50211864866466475700185501424, 8.073022984479922893785274081864, 9.266471151092426764479241198716, 9.863825680466408809676106446823
