| L(s) = 1 | − 1.41·2-s + 2.00·4-s + 2.64i·7-s − 2.82·8-s − 3.78i·11-s + 14.9i·13-s − 3.74i·14-s + 4.00·16-s − 0.335·17-s + 29.8·19-s + 5.35i·22-s + 18.5·23-s − 21.1i·26-s + 5.29i·28-s − 11.1i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.377i·7-s − 0.353·8-s − 0.344i·11-s + 1.14i·13-s − 0.267i·14-s + 0.250·16-s − 0.0197·17-s + 1.57·19-s + 0.243i·22-s + 0.805·23-s − 0.813i·26-s + 0.188i·28-s − 0.384i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.533878656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533878656\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 + 3.78iT - 121T^{2} \) |
| 13 | \( 1 - 14.9iT - 169T^{2} \) |
| 17 | \( 1 + 0.335T + 289T^{2} \) |
| 19 | \( 1 - 29.8T + 361T^{2} \) |
| 23 | \( 1 - 18.5T + 529T^{2} \) |
| 29 | \( 1 + 11.1iT - 841T^{2} \) |
| 31 | \( 1 - 0.603T + 961T^{2} \) |
| 37 | \( 1 + 41.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 35.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 50.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 44.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 16.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 17.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 76.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 63.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 60.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 129.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 15.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 59.4iT - 9.40e3T^{2} \) |
| 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
−8.795597923687872629390049302685, −7.75259047082901236742179656682, −7.26377957946448635327993633014, −6.39889130107595536501403899876, −5.65834033737072239599390632295, −4.79054709550708129711265961176, −3.69971097471159343181570423544, −2.79277545053777322948847131632, −1.81271236961421044407753933884, −0.75947216105064590715339970631,
0.64308938275513319163391889505, 1.48725029235610153031087252868, 2.84367122107741440026701483570, 3.41015134611795392220691098265, 4.71470562803428994909307648066, 5.41764915056958345951884868481, 6.32964581845269764666048172523, 7.20163730549416756131811381713, 7.69919500744322151241881558472, 8.379173852483069814111979000551
