L(s) = 1 | + (1.28 + 0.599i)2-s + 0.936·3-s + (1.28 + 1.53i)4-s − i·5-s + (1.19 + 0.561i)6-s + (−2.60 − 0.468i)7-s + (0.719 + 2.73i)8-s − 2.12·9-s + (0.599 − 1.28i)10-s − 2.39i·11-s + (1.19 + 1.43i)12-s − 2i·13-s + (−3.05 − 2.16i)14-s − 0.936i·15-s + (−0.719 + 3.93i)16-s + 7.12i·17-s + ⋯ |
L(s) = 1 | + (0.905 + 0.424i)2-s + 0.540·3-s + (0.640 + 0.768i)4-s − 0.447i·5-s + (0.489 + 0.229i)6-s + (−0.984 − 0.176i)7-s + (0.254 + 0.967i)8-s − 0.707·9-s + (0.189 − 0.405i)10-s − 0.723i·11-s + (0.346 + 0.415i)12-s − 0.554i·13-s + (−0.816 − 0.577i)14-s − 0.241i·15-s + (−0.179 + 0.983i)16-s + 1.72i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77092 + 0.468353i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77092 + 0.468353i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.28 - 0.599i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (2.60 + 0.468i)T \) |
good | 3 | \( 1 - 0.936T + 3T^{2} \) |
| 11 | \( 1 + 2.39iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 7.12iT - 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 + 5.73iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 7.12iT - 41T^{2} \) |
| 43 | \( 1 - 7.60iT - 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 14.2iT - 67T^{2} \) |
| 71 | \( 1 + 6.14iT - 71T^{2} \) |
| 73 | \( 1 - 9.36iT - 73T^{2} \) |
| 79 | \( 1 - 4.27iT - 79T^{2} \) |
| 83 | \( 1 - 0.936T + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 7.12iT - 97T^{2} \) |
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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
−13.21199995395032955297357356979, −12.67587225728130501601433445471, −11.46868064180587207923986383908, −10.23354193736687729885812485097, −8.692895077291286487287511184938, −8.036025060601533908303159213417, −6.46831283430676715366060310530, −5.61147730785555401941343622108, −3.93193478839480476454931162721, −2.85398824592260804443183247156,
2.52697532430818881695725016538, 3.49272063160832446146007279785, 5.13302341777082330521003232403, 6.45518617667683292029523023616, 7.45113572907467101968944943252, 9.336047205428287307280005866484, 9.894828267619128218742900518243, 11.42765847871728283033085848812, 11.98174159820559750101520720158, 13.35715193452817719441512953025