L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−1.86 − 1.23i)5-s − 0.999·6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.23 − 0.133i)10-s + (−1 + 1.73i)11-s + (−0.866 + 0.499i)12-s − 6i·13-s + (1 + 2i)15-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (0.866 + 0.499i)18-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.834 − 0.550i)5-s − 0.408·6-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.705 − 0.0423i)10-s + (−0.301 + 0.522i)11-s + (−0.249 + 0.144i)12-s − 1.66i·13-s + (0.258 + 0.516i)15-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + (0.204 + 0.117i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3461534629\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3461534629\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (1.86 + 1.23i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.46 - 2i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.73 - i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + (6.92 - 4i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.983666210349929925631776772434, −7.988506077597129856569704787717, −7.44195725748759341684006553207, −6.44791700748900122919109735561, −5.35390028600861272194413779920, −4.95409692887485365756739087212, −3.86284486703422862186272816595, −2.93059631107910561109890829733, −1.50163881377864309130114945278, −0.11706839766438174851007101220,
2.11620679027437038544633337597, 3.44154165933748422118529170494, 4.16879763376974909352460863865, 4.86998513912611385227291795502, 6.09714314901944488986813745170, 6.57752224739681717010977316784, 7.44196136654135656688954148220, 8.253586697818857743960737504628, 9.144269045424237678688770730300, 10.12999817863397011436770378594