L(s) = 1 | − 1.95i·3-s − 2.18·7-s + 5.19·9-s − 9.50·11-s + 20.4i·13-s − 6.15·17-s + (−9.68 + 16.3i)19-s + 4.26i·21-s + 10.9·23-s − 27.6i·27-s − 28.4i·29-s − 24.8i·31-s + 18.5i·33-s − 37.4i·37-s + 39.9·39-s + ⋯ |
L(s) = 1 | − 0.650i·3-s − 0.312·7-s + 0.577·9-s − 0.864·11-s + 1.57i·13-s − 0.362·17-s + (−0.509 + 0.860i)19-s + 0.203i·21-s + 0.475·23-s − 1.02i·27-s − 0.980i·29-s − 0.800i·31-s + 0.562i·33-s − 1.01i·37-s + 1.02·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.167576884\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167576884\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (9.68 - 16.3i)T \) |
good | 3 | \( 1 + 1.95iT - 9T^{2} \) |
| 7 | \( 1 + 2.18T + 49T^{2} \) |
| 11 | \( 1 + 9.50T + 121T^{2} \) |
| 13 | \( 1 - 20.4iT - 169T^{2} \) |
| 17 | \( 1 + 6.15T + 289T^{2} \) |
| 23 | \( 1 - 10.9T + 529T^{2} \) |
| 29 | \( 1 + 28.4iT - 841T^{2} \) |
| 31 | \( 1 + 24.8iT - 961T^{2} \) |
| 37 | \( 1 + 37.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 81.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 70.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 46.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 32.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 94.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 69.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 40.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 102.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 73.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 91.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 57.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 84.3iT - 9.40e3T^{2} \) |
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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
−8.869147819345493887898357346500, −7.69658902938263963062005008569, −7.34327098788471374455821612050, −6.40446821349518206528237636085, −5.79140497979208318562200022635, −4.50053743350812648342279981709, −3.92035974861358571189462686734, −2.44354807429714200617676007888, −1.77879590956963485126541972100, −0.32512441430490456808850636979,
1.08717804670119143486319717827, 2.70640427071815480877158489621, 3.31108259948404130654818723354, 4.57337071528397092916958583059, 5.04983406167677129031637417154, 6.03590842277732716484646375521, 6.99234413708341719200304354924, 7.74742700897094438579557871967, 8.575980161034994359608057429777, 9.374427396973979460481765972795