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
−13.35715193452817719441512953025, −11.98174159820559750101520720158, −11.42765847871728283033085848812, −9.894828267619128218742900518243, −9.336047205428287307280005866484, −7.45113572907467101968944943252, −6.45518617667683292029523023616, −5.13302341777082330521003232403, −3.49272063160832446146007279785, −2.52697532430818881695725016538,
2.85398824592260804443183247156, 3.93193478839480476454931162721, 5.61147730785555401941343622108, 6.46831283430676715366060310530, 8.036025060601533908303159213417, 8.692895077291286487287511184938, 10.23354193736687729885812485097, 11.46868064180587207923986383908, 12.67587225728130501601433445471, 13.21199995395032955297357356979