| L(s) = 1 | − 2.09·3-s + 6.85i·7-s − 4.62·9-s + (−10.3 − 3.61i)11-s + 8.70i·13-s − 30.8i·17-s − 13.4i·19-s − 14.3i·21-s − 8.15·23-s + 28.4·27-s + 32.9i·29-s − 33.0·31-s + (21.7 + 7.55i)33-s + 58.3·37-s − 18.2i·39-s + ⋯ |
| L(s) = 1 | − 0.697·3-s + 0.978i·7-s − 0.513·9-s + (−0.944 − 0.328i)11-s + 0.669i·13-s − 1.81i·17-s − 0.708i·19-s − 0.682i·21-s − 0.354·23-s + 1.05·27-s + 1.13i·29-s − 1.06·31-s + (0.658 + 0.228i)33-s + 1.57·37-s − 0.466i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9654770388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9654770388\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (10.3 + 3.61i)T \) |
| good | 3 | \( 1 + 2.09T + 9T^{2} \) |
| 7 | \( 1 - 6.85iT - 49T^{2} \) |
| 13 | \( 1 - 8.70iT - 169T^{2} \) |
| 17 | \( 1 + 30.8iT - 289T^{2} \) |
| 19 | \( 1 + 13.4iT - 361T^{2} \) |
| 23 | \( 1 + 8.15T + 529T^{2} \) |
| 29 | \( 1 - 32.9iT - 841T^{2} \) |
| 31 | \( 1 + 33.0T + 961T^{2} \) |
| 37 | \( 1 - 58.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 80.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 38.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 0.654T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 23.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 70.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 56.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 46.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 137.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 154.T + 9.40e3T^{2} \) |
| 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
−9.426236030168465386858653384551, −8.981598088688921805604239462455, −7.973109207627298790199551158028, −7.02094552235772968165930093442, −6.10233440936881478161143538559, −5.30056468469963460622053781942, −4.77206898501222271958298069031, −3.10427641684467057508633716548, −2.33282896416878340964112733630, −0.50100307801364174721336801388,
0.71364416329189888696823175943, 2.23665485641530791995674368576, 3.62522300448168248427422971016, 4.45742640069418775752677248325, 5.73489840477223028012149574410, 5.98882135332124128873809478442, 7.34415451673055924447110920893, 7.913759657245033003788007068540, 8.773434509811502952611854230327, 10.19191715208605764023139502902
