L(s) = 1 | + 2.08i·2-s − 2.34·4-s + (−1.65 + 2.86i)5-s − 0.717i·8-s + (−5.96 − 3.44i)10-s + (2.30 − 1.33i)11-s + (2.11 − 1.21i)13-s − 3.19·16-s + (−3.59 + 6.21i)17-s + (−4.24 + 2.45i)19-s + (3.87 − 6.70i)20-s + (2.77 + 4.80i)22-s + (−4.32 − 2.49i)23-s + (−2.96 − 5.12i)25-s + (2.54 + 4.40i)26-s + ⋯ |
L(s) = 1 | + 1.47i·2-s − 1.17·4-s + (−0.738 + 1.27i)5-s − 0.253i·8-s + (−1.88 − 1.08i)10-s + (0.694 − 0.401i)11-s + (0.585 − 0.338i)13-s − 0.798·16-s + (−0.870 + 1.50i)17-s + (−0.974 + 0.562i)19-s + (0.866 − 1.50i)20-s + (0.591 + 1.02i)22-s + (−0.901 − 0.520i)23-s + (−0.592 − 1.02i)25-s + (0.498 + 0.863i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6467028604\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6467028604\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.08iT - 2T^{2} \) |
| 5 | \( 1 + (1.65 - 2.86i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.30 + 1.33i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.11 + 1.21i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.59 - 6.21i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.24 - 2.45i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.32 + 2.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.50 + 3.17i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.66iT - 31T^{2} \) |
| 37 | \( 1 + (-0.844 - 1.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.553 + 0.958i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.93 + 5.08i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 + (-8.94 - 5.16i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 5.13T + 59T^{2} \) |
| 61 | \( 1 + 5.13iT - 61T^{2} \) |
| 67 | \( 1 - 8.33T + 67T^{2} \) |
| 71 | \( 1 - 2.07iT - 71T^{2} \) |
| 73 | \( 1 + (6.94 + 4.00i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 + (1.04 - 1.80i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.541 - 0.937i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.47 + 5.46i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.40430667591151254251563797255, −9.002524478545535249327012534926, −8.346818304753300546156119590142, −7.76693113348590289093517952886, −6.85107055285558915359371987494, −6.29159927453797915270160626952, −5.77752504653491183621644504755, −4.18272311340416392055253762913, −3.74467886063448819447631165949, −2.20635164051639151798634641998,
0.27107194140133349599047742223, 1.43362532893993916461560825063, 2.49737624402106799840941268206, 3.93905669692636394436128597025, 4.23973058765244457325686762925, 5.18363815457661045395773116426, 6.57515325147884549279945809780, 7.51755477146237272430780175321, 8.687468669219100559319969924830, 9.120239609274353898564499816494