L(s) = 1 | + (0.596 + 0.433i)2-s + (−0.868 + 2.67i)3-s + (−0.449 − 1.38i)4-s + (0.809 − 0.587i)5-s + (−1.67 + 1.21i)6-s + (0.318 + 0.980i)7-s + (0.787 − 2.42i)8-s + (−3.96 − 2.88i)9-s + 0.737·10-s + (1.93 − 2.69i)11-s + 4.09·12-s + (−2.79 − 2.02i)13-s + (−0.235 + 0.723i)14-s + (0.868 + 2.67i)15-s + (−0.834 + 0.606i)16-s + (−1.94 + 1.40i)17-s + ⋯ |
L(s) = 1 | + (0.421 + 0.306i)2-s + (−0.501 + 1.54i)3-s + (−0.224 − 0.692i)4-s + (0.361 − 0.262i)5-s + (−0.684 + 0.497i)6-s + (0.120 + 0.370i)7-s + (0.278 − 0.857i)8-s + (−1.32 − 0.961i)9-s + 0.233·10-s + (0.583 − 0.811i)11-s + 1.18·12-s + (−0.773 − 0.562i)13-s + (−0.0628 + 0.193i)14-s + (0.224 + 0.690i)15-s + (−0.208 + 0.151i)16-s + (−0.470 + 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.796395 + 0.420186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.796395 + 0.420186i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (-1.93 + 2.69i)T \) |
good | 2 | \( 1 + (-0.596 - 0.433i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.868 - 2.67i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-0.318 - 0.980i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (2.79 + 2.02i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.94 - 1.40i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.36 - 7.29i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 2.45T + 23T^{2} \) |
| 29 | \( 1 + (1.83 + 5.66i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.98 - 2.16i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.84 - 5.66i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.21 + 3.74i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 7.64T + 43T^{2} \) |
| 47 | \( 1 + (-1.80 + 5.55i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.58 - 6.96i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.910 - 2.80i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.00 - 1.45i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 + (1.63 - 1.18i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.255 + 0.785i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.77 - 7.09i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (1.30 - 0.946i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 8.16T + 89T^{2} \) |
| 97 | \( 1 + (1.97 + 1.43i)T + (29.9 + 92.2i)T^{2} \) |
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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
−15.26316081571501091390682351960, −14.81654747253510764756256985637, −13.56925990542647407919013103904, −11.97701002392100713111873507865, −10.56931121538238887032405184149, −9.888524781409451832097569030107, −8.727229894635258941157291925049, −6.12051626079621319602845311024, −5.28116635919281508337924941740, −4.03154735665659843412821333460,
2.31346873743097894295491064814, 4.75197818114324137406326172222, 6.75194129837842569168796799919, 7.40963601379384653372287823625, 9.071398378020895719399577538642, 11.11746631921388414969948292222, 11.97909322681601117888362037108, 12.93088582731995426365198531894, 13.59370688928729702329316660612, 14.70276550900625826788232814992