L(s) = 1 | + (1.69 + 1.06i)2-s + (0.680 − 0.733i)3-s + (1.29 + 2.69i)4-s + (1.93 − 0.517i)6-s + (−0.446 + 3.96i)8-s + (−0.0747 − 0.997i)9-s + (2.86 + 0.882i)12-s + (−1.14 − 0.914i)13-s + (−3.10 + 3.88i)16-s + (0.933 − 1.76i)18-s + (−0.974 − 0.222i)23-s + (2.60 + 3.02i)24-s + (−0.900 + 0.433i)25-s + (−0.967 − 2.76i)26-s + (−0.781 − 0.623i)27-s + ⋯ |
L(s) = 1 | + (1.69 + 1.06i)2-s + (0.680 − 0.733i)3-s + (1.29 + 2.69i)4-s + (1.93 − 0.517i)6-s + (−0.446 + 3.96i)8-s + (−0.0747 − 0.997i)9-s + (2.86 + 0.882i)12-s + (−1.14 − 0.914i)13-s + (−3.10 + 3.88i)16-s + (0.933 − 1.76i)18-s + (−0.974 − 0.222i)23-s + (2.60 + 3.02i)24-s + (−0.900 + 0.433i)25-s + (−0.967 − 2.76i)26-s + (−0.781 − 0.623i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.446708845\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.446708845\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.680 + 0.733i)T \) |
| 23 | \( 1 + (0.974 + 0.222i)T \) |
| 29 | \( 1 + (-0.563 + 0.826i)T \) |
good | 2 | \( 1 + (-1.69 - 1.06i)T + (0.433 + 0.900i)T^{2} \) |
| 5 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 7 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 11 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 13 | \( 1 + (1.14 + 0.914i)T + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 31 | \( 1 + (-1.02 + 1.63i)T + (-0.433 - 0.900i)T^{2} \) |
| 37 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 41 | \( 1 + (-0.565 - 0.565i)T + iT^{2} \) |
| 43 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 47 | \( 1 + (-0.369 + 0.0416i)T + (0.974 - 0.222i)T^{2} \) |
| 53 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 - 1.80iT - T^{2} \) |
| 61 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 67 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 71 | \( 1 + (0.367 - 0.460i)T + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (0.806 + 1.28i)T + (-0.433 + 0.900i)T^{2} \) |
| 79 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 83 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 97 | \( 1 + (0.781 + 0.623i)T^{2} \) |
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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
−9.155858153464336611362117221457, −8.022653821931747072885079396314, −7.82076184945436871951635080403, −7.15321135190117377133243291654, −6.10744160793023562204509443934, −5.81038789447162561491933846494, −4.58431077882121011618891905479, −3.92546505968028078171222820743, −2.81757457939640707320838845978, −2.31565470252632024713202170538,
1.79633969672301374502604461980, 2.53862250023522702512304287767, 3.41880989042496094579119782588, 4.24014837004915835384035470083, 4.79615253858860492259271741548, 5.54787844685853824725326265237, 6.57215665115214757650666216914, 7.37342964556104511925553611990, 8.701059012611473538275083391363, 9.680094753839869001170338855382