L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s + (0.749 − 0.432i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.151 + 0.0874i)11-s − 0.999·12-s + (−3.34 + 1.35i)13-s + 0.865·14-s + (−0.5 + 0.866i)16-s + (−4.08 − 7.08i)17-s − 0.999i·18-s + (−5.20 + 3.00i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.353 + 0.204i)6-s + (0.283 − 0.163i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.0456 + 0.0263i)11-s − 0.288·12-s + (−0.926 + 0.375i)13-s + 0.231·14-s + (−0.125 + 0.216i)16-s + (−0.991 − 1.71i)17-s − 0.235i·18-s + (−1.19 + 0.689i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6291328622\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6291328622\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.34 - 1.35i)T \) |
good | 7 | \( 1 + (-0.749 + 0.432i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.151 - 0.0874i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (4.08 + 7.08i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.20 - 3.00i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.45 - 2.52i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.24 - 5.62i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.95iT - 31T^{2} \) |
| 37 | \( 1 + (-1.52 - 0.879i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.08 + 4.08i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.58 + 7.94i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 11.9iT - 47T^{2} \) |
| 53 | \( 1 + 2.48T + 53T^{2} \) |
| 59 | \( 1 + (-6.09 + 3.51i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.98 - 6.90i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.36 + 1.36i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.2 + 7.08i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 12.8iT - 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 - 0.139iT - 83T^{2} \) |
| 89 | \( 1 + (-11.3 - 6.56i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.48 - 4.32i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.511017493222334882473410748481, −8.924696384578938236272873566831, −7.945026201049029660954384266396, −7.03106633708668702895408196865, −6.56215254053438823985416100834, −5.35267446153942934785501576448, −4.86254307313987565199508721689, −4.09318927449226179811658622905, −3.05755690870974486394960422417, −1.91994303948582243142866101005,
0.17042252445382834671199520316, 1.89841667845188042641058893477, 2.45201082567212062341054699375, 3.89130556026241405098904044224, 4.59324452137199814911546590839, 5.52159802280319095714890040334, 6.35396197258126392651014430678, 6.88233097319216569471442340665, 8.090222673376280842532699243856, 8.505172910072638807808033838838