L(s) = 1 | + i·2-s + (0.352 + 1.69i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−1.69 + 0.352i)6-s + (2.55 − 0.696i)7-s − i·8-s + (−2.75 + 1.19i)9-s + (−0.866 − 0.5i)10-s + (1.26 − 0.732i)11-s + (−0.352 − 1.69i)12-s + (−6.03 + 3.48i)13-s + (0.696 + 2.55i)14-s + (−1.64 − 0.542i)15-s + 16-s + (−3.30 + 5.72i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.203 + 0.979i)3-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.692 + 0.143i)6-s + (0.964 − 0.263i)7-s − 0.353i·8-s + (−0.917 + 0.398i)9-s + (−0.273 − 0.158i)10-s + (0.382 − 0.220i)11-s + (−0.101 − 0.489i)12-s + (−1.67 + 0.966i)13-s + (0.186 + 0.682i)14-s + (−0.424 − 0.140i)15-s + 0.250·16-s + (−0.801 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0608679 - 1.16064i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0608679 - 1.16064i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.352 - 1.69i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.55 + 0.696i)T \) |
good | 11 | \( 1 + (-1.26 + 0.732i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.03 - 3.48i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.30 - 5.72i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.08 - 1.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.51 - 3.18i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.49 - 0.862i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.84iT - 31T^{2} \) |
| 37 | \( 1 + (2.75 + 4.77i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.632 - 1.09i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.24 - 5.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 5.39T + 47T^{2} \) |
| 53 | \( 1 + (-5.60 - 3.23i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 1.49T + 59T^{2} \) |
| 61 | \( 1 + 3.73iT - 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 14.2iT - 71T^{2} \) |
| 73 | \( 1 + (1.11 + 0.640i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 0.994T + 79T^{2} \) |
| 83 | \( 1 + (5.93 - 10.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.18 - 3.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.04 - 5.22i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.98096477574242540065023020370, −10.08407266176172731511915564899, −9.207880009623218811599645878730, −8.442650770414374623491027370400, −7.58571070905907137732043165004, −6.64736931192081203691661260604, −5.47437228583518390937145945939, −4.47895071592582540077019383532, −3.92894745379121573123718363131, −2.23980205277307343445599875490,
0.60849635856559628993597423979, 2.10624579523580455646000511360, 2.93852420686230266514052436445, 4.72492140164767055055349627020, 5.18905403888987464604196764199, 6.84552295608655076110764177471, 7.51777113908411704045570938801, 8.593518828773221686274291561771, 9.023336405854107015970116071999, 10.27419606689091581218315410415