L(s) = 1 | + (1.73 − 2.44i)3-s − 2.82i·5-s − 10.3·7-s + (−2.99 − 8.48i)9-s − 14.6i·11-s + 6·13-s + (−6.92 − 4.89i)15-s + 22.6i·17-s − 10.3·19-s + (−18 + 25.4i)21-s − 29.3i·23-s + 17·25-s + (−25.9 − 7.34i)27-s − 31.1i·29-s + 31.1·31-s + ⋯ |
L(s) = 1 | + (0.577 − 0.816i)3-s − 0.565i·5-s − 1.48·7-s + (−0.333 − 0.942i)9-s − 1.33i·11-s + 0.461·13-s + (−0.461 − 0.326i)15-s + 1.33i·17-s − 0.546·19-s + (−0.857 + 1.21i)21-s − 1.27i·23-s + 0.680·25-s + (−0.962 − 0.272i)27-s − 1.07i·29-s + 1.00·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.613442 - 1.18507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.613442 - 1.18507i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.73 + 2.44i)T \) |
good | 5 | \( 1 + 2.82iT - 25T^{2} \) |
| 7 | \( 1 + 10.3T + 49T^{2} \) |
| 11 | \( 1 + 14.6iT - 121T^{2} \) |
| 13 | \( 1 - 6T + 169T^{2} \) |
| 17 | \( 1 - 22.6iT - 289T^{2} \) |
| 19 | \( 1 + 10.3T + 361T^{2} \) |
| 23 | \( 1 + 29.3iT - 529T^{2} \) |
| 29 | \( 1 + 31.1iT - 841T^{2} \) |
| 31 | \( 1 - 31.1T + 961T^{2} \) |
| 37 | \( 1 - 38T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.65iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 58.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 14.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 14.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 22T + 3.72e3T^{2} \) |
| 67 | \( 1 - 114.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 29.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30T + 5.32e3T^{2} \) |
| 79 | \( 1 - 31.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 73.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 5.65iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 90T + 9.40e3T^{2} \) |
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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
−12.39358998129373484541994972047, −11.03472942988146290064782667668, −9.808908919737260454698142727809, −8.692018812970710695813805880800, −8.189674277927554316364104022505, −6.48029169248835227040527329569, −6.09999779994752604948496080480, −3.92772572483879986084655211127, −2.75157826694989290057115602843, −0.72401904427738249545463080342,
2.63146738052183941462554433015, 3.63447412582824223831344670747, 4.99712288124420771931009101710, 6.55217722705508720723459399003, 7.45436655754535338171488764119, 8.978208069888724654404750193979, 9.734421673167430953856171071532, 10.35446439392920892445786216936, 11.56247834361556696060022006960, 12.81754239359315597795646593055