L(s) = 1 | + 2.23i·2-s − 3.00·4-s + 3.87·5-s + (−2 + 1.73i)7-s − 2.23i·8-s + 8.66i·10-s + 2.23i·11-s − 3.46i·13-s + (−3.87 − 4.47i)14-s − 0.999·16-s − 5.19i·19-s − 11.6·20-s − 5.00·22-s + 2.23i·23-s + 10.0·25-s + 7.74·26-s + ⋯ |
L(s) = 1 | + 1.58i·2-s − 1.50·4-s + 1.73·5-s + (−0.755 + 0.654i)7-s − 0.790i·8-s + 2.73i·10-s + 0.674i·11-s − 0.960i·13-s + (−1.03 − 1.19i)14-s − 0.249·16-s − 1.19i·19-s − 2.59·20-s − 1.06·22-s + 0.466i·23-s + 2.00·25-s + 1.51·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.547839 + 1.19916i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.547839 + 1.19916i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 11 | \( 1 - 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 2.23iT - 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 3.87T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 7.74T + 47T^{2} \) |
| 53 | \( 1 - 8.94iT - 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 11.1iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−13.35078624594296651534536574375, −12.41542773185722704789239987634, −10.58231632012419085804117972913, −9.457769710290539094557814215942, −9.041912193689631135262666294743, −7.58381904159027412402719836443, −6.46441248412202969167806553322, −5.82526903388341409099158626797, −4.95738757290599019952705139486, −2.54379477291757560647864572511,
1.46341883486654928671535527742, 2.76691047005576429478913120071, 4.09457980771662169030647271265, 5.72926167363061150772777409765, 6.78709535216385006703237089919, 8.836656498878657218387275522995, 9.629390018444784495463203215370, 10.27248587092429499816909287478, 11.00897137844238312897365585340, 12.29820213580494058706517612021