L(s) = 1 | + (0.337 − 1.37i)2-s + (−1.77 − 0.928i)4-s + (−1.32 + 3.19i)5-s + (−2.32 + 2.32i)7-s + (−1.87 + 2.11i)8-s + (3.93 + 2.89i)10-s + (−1.47 + 3.55i)11-s + (4.49 − 1.86i)13-s + (2.41 + 3.98i)14-s + (2.27 + 3.28i)16-s − 4.93·17-s + (−1.98 − 4.79i)19-s + (5.30 − 4.42i)20-s + (4.37 + 3.21i)22-s + (−1.08 + 1.08i)23-s + ⋯ |
L(s) = 1 | + (0.238 − 0.971i)2-s + (−0.885 − 0.464i)4-s + (−0.591 + 1.42i)5-s + (−0.880 + 0.880i)7-s + (−0.662 + 0.749i)8-s + (1.24 + 0.915i)10-s + (−0.443 + 1.07i)11-s + (1.24 − 0.516i)13-s + (0.644 + 1.06i)14-s + (0.569 + 0.822i)16-s − 1.19·17-s + (−0.455 − 1.09i)19-s + (1.18 − 0.990i)20-s + (0.933 + 0.686i)22-s + (−0.227 + 0.227i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.624283 + 0.416311i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.624283 + 0.416311i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.337 + 1.37i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.32 - 3.19i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (2.32 - 2.32i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.47 - 3.55i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-4.49 + 1.86i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 + (1.98 + 4.79i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.08 - 1.08i)T - 23iT^{2} \) |
| 29 | \( 1 + (-3.43 + 1.42i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 8.82iT - 31T^{2} \) |
| 37 | \( 1 + (-1.94 - 0.804i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-5.87 - 5.87i)T + 41iT^{2} \) |
| 43 | \( 1 + (2.44 + 1.01i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 1.61iT - 47T^{2} \) |
| 53 | \( 1 + (-5.62 - 2.32i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (7.67 + 3.17i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-3.16 - 7.65i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-3.31 + 1.37i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (2.13 + 2.13i)T + 71iT^{2} \) |
| 73 | \( 1 + (1.81 - 1.81i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 + (1.04 - 0.431i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.708 + 0.708i)T - 89iT^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−11.88431378593404502174830577241, −10.97677507768550494827931021002, −10.44878069308568438984242907333, −9.368944838075084701808465672304, −8.395798659729350437019062220789, −6.91769615649059870334879938733, −6.06119302632428565756191755257, −4.51881531103238109129013976035, −3.22095140096674258013779294297, −2.46843272947750395743286637086,
0.51868703733119623542674919936, 3.75017012858800716707488841683, 4.31929895907815831776106175530, 5.74922285286867964782839710563, 6.57675514686944741720198643868, 7.923463028780076932398557283070, 8.558498016664473227692928066651, 9.330560596689832370711718713783, 10.67041260923716067890661834558, 11.87601662936594198461172292658