L(s) = 1 | + (−0.535 + 1.30i)2-s + (−1.42 − 1.40i)4-s + (−1.93 + 1.11i)5-s + (−0.5 − 2.59i)7-s + (2.59 − 1.11i)8-s + (−0.427 − 3.13i)10-s + (1.93 + 1.11i)11-s + (3.66 + 0.736i)14-s + (0.0729 + 3.99i)16-s + (1.73 − 3i)17-s + (6.70 − 3.87i)19-s + (4.33 + 1.11i)20-s + (−2.5 + 1.93i)22-s + (3.46 + 6i)23-s + (−2.92 + 4.40i)28-s − 2.23i·29-s + ⋯ |
L(s) = 1 | + (−0.378 + 0.925i)2-s + (−0.713 − 0.700i)4-s + (−0.866 + 0.499i)5-s + (−0.188 − 0.981i)7-s + (0.918 − 0.395i)8-s + (−0.135 − 0.990i)10-s + (0.583 + 0.337i)11-s + (0.980 + 0.196i)14-s + (0.0182 + 0.999i)16-s + (0.420 − 0.727i)17-s + (1.53 − 0.888i)19-s + (0.968 + 0.249i)20-s + (−0.533 + 0.412i)22-s + (0.722 + 1.25i)23-s + (−0.553 + 0.833i)28-s − 0.415i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.919235 + 0.251008i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.919235 + 0.251008i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.535 - 1.30i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 5 | \( 1 + (1.93 - 1.11i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.93 - 1.11i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (-1.73 + 3i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.70 + 3.87i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.46 - 6i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.23iT - 29T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.70 + 3.87i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + (1.73 + 3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.68 + 5.59i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.93 - 1.11i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.70 + 3.87i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.70 - 3.87i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 + (6.92 + 12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−11.08707918754753521442498279826, −9.684278348224833984841533655823, −9.430787580000023680403832827787, −7.894762801487939336625358360542, −7.35882326520100336473065977923, −6.81593480233986792169031447544, −5.46260436227634966840448415583, −4.35265963019640119225915143568, −3.35256571302802823548628651644, −0.877941589925355588623482634574,
1.12965146234047735711814096702, 2.82092461034985427421944540261, 3.82392804144644238671811658166, 4.89767286388241139039081893135, 6.11814265945284479968454210514, 7.66222341911397303714812069541, 8.344430479337587471424851803297, 9.105233433940507687400332115646, 9.888609509982890140732495887142, 11.02318389468497229535494925468