L(s) = 1 | + (−0.623 + 1.78i)2-s + (−0.222 − 0.974i)3-s + (−2.00 − 1.59i)4-s + (1.87 + 0.211i)6-s + (2.49 − 1.57i)8-s + (−0.900 + 0.433i)9-s + (−1.11 + 2.30i)12-s + (−1.75 + 0.400i)13-s + (0.669 + 2.93i)16-s + (−0.211 − 1.87i)18-s + (−0.433 + 0.900i)23-s + (−2.08 − 2.08i)24-s + (0.623 − 0.781i)25-s + (0.380 − 3.38i)26-s + (0.623 + 0.781i)27-s + ⋯ |
L(s) = 1 | + (−0.623 + 1.78i)2-s + (−0.222 − 0.974i)3-s + (−2.00 − 1.59i)4-s + (1.87 + 0.211i)6-s + (2.49 − 1.57i)8-s + (−0.900 + 0.433i)9-s + (−1.11 + 2.30i)12-s + (−1.75 + 0.400i)13-s + (0.669 + 2.93i)16-s + (−0.211 − 1.87i)18-s + (−0.433 + 0.900i)23-s + (−2.08 − 2.08i)24-s + (0.623 − 0.781i)25-s + (0.380 − 3.38i)26-s + (0.623 + 0.781i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4522588296\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4522588296\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.222 + 0.974i)T \) |
| 23 | \( 1 + (0.433 - 0.900i)T \) |
| 29 | \( 1 + (-0.781 - 0.623i)T \) |
good | 2 | \( 1 + (0.623 - 1.78i)T + (-0.781 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 13 | \( 1 + (1.75 - 0.400i)T + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 31 | \( 1 + (-1.33 - 0.467i)T + (0.781 + 0.623i)T^{2} \) |
| 37 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 41 | \( 1 + (-0.752 - 0.752i)T + iT^{2} \) |
| 43 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 47 | \( 1 + (1.05 - 1.68i)T + (-0.433 - 0.900i)T^{2} \) |
| 53 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 - 1.24iT - T^{2} \) |
| 61 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 67 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (0.193 + 0.846i)T + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-1.87 + 0.656i)T + (0.781 - 0.623i)T^{2} \) |
| 79 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 83 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 97 | \( 1 + (0.974 - 0.222i)T^{2} \) |
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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
−9.404769899563263248644833179807, −8.543146676286509077783810512755, −7.81596627801746528099906010979, −7.40040542694339893679835513968, −6.55792127718321128352289933756, −6.15361131031997115697961724863, −5.04530858755335073485963135015, −4.61893769783894818744984020336, −2.69325925309115995567696262578, −1.17947667341555740482404024951,
0.47403144761599708117908831176, 2.27338083380008952489515309338, 2.90724172506648742573410154711, 3.89505425516310588510056790561, 4.68798217795589366467592404820, 5.29453146117627871355035786781, 6.79322644173615937281670663981, 8.057157376613406506727143770895, 8.516529720050788545303150837436, 9.579871726088194199343541070784