L(s) = 1 | + (−1.38 − 0.273i)2-s + (0.758 − 1.55i)3-s + (1.85 + 0.758i)4-s + (−1.47 + 1.95i)6-s − 3.56·7-s + (−2.36 − 1.55i)8-s + (−1.85 − 2.36i)9-s − 4.20·11-s + (2.58 − 2.30i)12-s − 2.70i·13-s + (4.94 + 0.973i)14-s + (2.85 + 2.80i)16-s + 0.828·17-s + (1.92 + 3.78i)18-s − 5.07i·19-s + ⋯ |
L(s) = 1 | + (−0.981 − 0.193i)2-s + (0.437 − 0.899i)3-s + (0.925 + 0.379i)4-s + (−0.603 + 0.797i)6-s − 1.34·7-s + (−0.834 − 0.550i)8-s + (−0.616 − 0.787i)9-s − 1.26·11-s + (0.745 − 0.666i)12-s − 0.749i·13-s + (1.32 + 0.260i)14-s + (0.712 + 0.701i)16-s + 0.200·17-s + (0.453 + 0.891i)18-s − 1.16i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0642412 - 0.481199i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0642412 - 0.481199i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.38 + 0.273i)T \) |
| 3 | \( 1 + (-0.758 + 1.55i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 + 2.70iT - 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 5.07iT - 19T^{2} \) |
| 23 | \( 1 + 1.09iT - 23T^{2} \) |
| 29 | \( 1 - 5.55iT - 29T^{2} \) |
| 31 | \( 1 + 6.59iT - 31T^{2} \) |
| 37 | \( 1 - 5.40iT - 37T^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 - 0.531T + 43T^{2} \) |
| 47 | \( 1 - 6.22iT - 47T^{2} \) |
| 53 | \( 1 - 5.55T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.701T + 61T^{2} \) |
| 67 | \( 1 - 2.04T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 7.70iT - 73T^{2} \) |
| 79 | \( 1 - 7.12iT - 79T^{2} \) |
| 83 | \( 1 + 3.11iT - 83T^{2} \) |
| 89 | \( 1 + 4.72iT - 89T^{2} \) |
| 97 | \( 1 + 8.10iT - 97T^{2} \) |
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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
−11.17815610293505655504300211323, −10.23056532439529955873690070430, −9.363595794065071824180535727861, −8.453282004325068173905884621237, −7.52073581367453304651307977949, −6.76720332419199900893908225419, −5.70642687021957176100572909443, −3.26097086463846921134032801587, −2.49672116770888570075259609020, −0.42493530527779318952078622417,
2.46468720516713949845520552896, 3.61191363385822460881960737790, 5.33743069404428223238295865131, 6.38916068399799801836951711949, 7.63202792984300529490609044169, 8.509433032856136101861773733624, 9.545698648613539973137511945410, 10.02697040986554562485617405418, 10.77871427867335976648301728142, 11.93425122382105795972320115058