L(s) = 1 | − 4.56·3-s + 5.73i·5-s + 2.64i·7-s + 11.8·9-s + 1.40·11-s − 19.0i·13-s − 26.1i·15-s − 32.2·17-s − 12.5·19-s − 12.0i·21-s − 15.8i·23-s − 7.86·25-s − 13.0·27-s − 3.29i·29-s − 22.6i·31-s + ⋯ |
L(s) = 1 | − 1.52·3-s + 1.14i·5-s + 0.377i·7-s + 1.31·9-s + 0.127·11-s − 1.46i·13-s − 1.74i·15-s − 1.89·17-s − 0.661·19-s − 0.575i·21-s − 0.690i·23-s − 0.314·25-s − 0.484·27-s − 0.113i·29-s − 0.731i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0926210 - 0.171085i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0926210 - 0.171085i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 + 4.56T + 9T^{2} \) |
| 5 | \( 1 - 5.73iT - 25T^{2} \) |
| 11 | \( 1 - 1.40T + 121T^{2} \) |
| 13 | \( 1 + 19.0iT - 169T^{2} \) |
| 17 | \( 1 + 32.2T + 289T^{2} \) |
| 19 | \( 1 + 12.5T + 361T^{2} \) |
| 23 | \( 1 + 15.8iT - 529T^{2} \) |
| 29 | \( 1 + 3.29iT - 841T^{2} \) |
| 31 | \( 1 + 22.6iT - 961T^{2} \) |
| 37 | \( 1 + 54.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 7.59T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 21.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 0.356iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 26.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 86.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 114.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 24.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 79.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 2.66T + 7.92e3T^{2} \) |
| 97 | \( 1 + 52.0T + 9.40e3T^{2} \) |
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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
−11.45733628691509536203140959124, −10.79140068578799233977449563015, −10.30924753755196143245844933768, −8.770700323855977081091246176926, −7.29682983973262612233178177770, −6.39299562918920332854867286191, −5.68512164585197556090120817138, −4.35430646577268367463356695720, −2.57297291506444161396090485057, −0.12962795379553296829796567929,
1.51183478155550317557141775287, 4.36719874430586989441543684022, 4.85660935491023157703044268636, 6.23741470442013812376068450893, 6.94406752730100100576652253844, 8.588748800958123551780917182151, 9.431762252987307922292043061178, 10.74090818118763758149071670301, 11.43211745405745317129325920659, 12.20524020423738395915277380706