| L(s) = 1 | + 2.16·3-s − 8.18i·7-s − 4.29·9-s + 3.30·11-s − 8.86·13-s − 10.2i·17-s + (−18.5 − 4.03i)19-s − 17.7i·21-s + 0.896i·23-s − 28.8·27-s + 17.9i·29-s + 40.8i·31-s + 7.16·33-s + 7.68·37-s − 19.2·39-s + ⋯ |
| L(s) = 1 | + 0.723·3-s − 1.16i·7-s − 0.477·9-s + 0.300·11-s − 0.681·13-s − 0.602i·17-s + (−0.977 − 0.212i)19-s − 0.845i·21-s + 0.0389i·23-s − 1.06·27-s + 0.617i·29-s + 1.31i·31-s + 0.217·33-s + 0.207·37-s − 0.492·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2341216661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2341216661\) |
| \(L(2)\) |
|
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 | \( 1 + (18.5 + 4.03i)T \) |
| good | 3 | \( 1 - 2.16T + 9T^{2} \) |
| 7 | \( 1 + 8.18iT - 49T^{2} \) |
| 11 | \( 1 - 3.30T + 121T^{2} \) |
| 13 | \( 1 + 8.86T + 169T^{2} \) |
| 17 | \( 1 + 10.2iT - 289T^{2} \) |
| 23 | \( 1 - 0.896iT - 529T^{2} \) |
| 29 | \( 1 - 17.9iT - 841T^{2} \) |
| 31 | \( 1 - 40.8iT - 961T^{2} \) |
| 37 | \( 1 - 7.68T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 25.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 12.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 55.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 105. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 14.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 2.89iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 75.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 86.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 52.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 39.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 172.T + 9.40e3T^{2} \) |
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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
−9.383796061539545602343244161860, −8.466702784342287251656338482893, −7.87224813407455766035946501409, −7.02733907445595811148888838118, −6.40072944207148815185409449705, −5.12176407084138494795804874753, −4.36591457894974878646749710244, −3.39563213207648588755649856371, −2.61799140444965611479892767132, −1.34069733161934238483076892545,
0.05048782377457342256015997853, 2.00072351412597789877082032321, 2.51324103021606219697971424844, 3.59265028851564557398761210646, 4.53003128278839703115932320130, 5.71900467180921040903192192518, 6.11158706120453151276800797027, 7.33240302518553350007792512148, 8.108755390806964608529282405088, 8.802169938853948004323015217111
