L(s) = 1 | − 2.49·3-s + (0.335 − 0.243i)5-s + (0.309 − 0.951i)7-s + 3.21·9-s + (−2.96 − 2.15i)11-s + (1.82 + 5.61i)13-s + (−0.835 + 0.606i)15-s + (−2.74 − 1.99i)17-s + (0.896 − 2.75i)19-s + (−0.770 + 2.37i)21-s + (−1.87 − 5.76i)23-s + (−1.49 + 4.59i)25-s − 0.524·27-s + (4.51 − 3.27i)29-s + (1.80 + 1.31i)31-s + ⋯ |
L(s) = 1 | − 1.43·3-s + (0.149 − 0.108i)5-s + (0.116 − 0.359i)7-s + 1.07·9-s + (−0.895 − 0.650i)11-s + (0.506 + 1.55i)13-s + (−0.215 + 0.156i)15-s + (−0.666 − 0.483i)17-s + (0.205 − 0.632i)19-s + (−0.168 + 0.517i)21-s + (−0.390 − 1.20i)23-s + (−0.298 + 0.918i)25-s − 0.100·27-s + (0.837 − 0.608i)29-s + (0.324 + 0.235i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3883535864\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3883535864\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (5.21 - 3.71i)T \) |
good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 + (-0.335 + 0.243i)T + (1.54 - 4.75i)T^{2} \) |
| 11 | \( 1 + (2.96 + 2.15i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.82 - 5.61i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.74 + 1.99i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.896 + 2.75i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.87 + 5.76i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.51 + 3.27i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.80 - 1.31i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.56 - 1.86i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-2.93 - 9.03i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-2.29 - 7.07i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (7.86 - 5.71i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.0716 + 0.220i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.19 - 9.82i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (9.04 - 6.56i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (3.21 + 2.33i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + 2.86T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + (2.10 - 6.48i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-11.7 + 8.51i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.32935200077460389979528630765, −9.310178362470632096642261652083, −8.479594261935599814190221885346, −7.36116669122793481084091819486, −6.46887433200620118880150883661, −6.00680386547698052494802148109, −4.82254196931541173251276797138, −4.41698019792117020230229668249, −2.77611811465413412057021161114, −1.20262544711054714880500540580,
0.22472562309755801783440905706, 1.87108249982408437847126942528, 3.30411039497061310453482974413, 4.63602829508312654540970162405, 5.50992168552351805718502133701, 5.87608296872573120054545425928, 6.88790046114137833805497136124, 7.85280711828038333961589406370, 8.609577474797080818277606013632, 10.00091799856893566547741131054