L(s) = 1 | − 2.82i·3-s − 2.82i·7-s − 5.00·9-s + 5.65·11-s − 2i·13-s − 2i·17-s − 8.00·21-s − 2.82i·23-s + 5.65i·27-s − 6·29-s − 5.65·31-s − 16.0i·33-s + 10i·37-s − 5.65·39-s + 2·41-s + ⋯ |
L(s) = 1 | − 1.63i·3-s − 1.06i·7-s − 1.66·9-s + 1.70·11-s − 0.554i·13-s − 0.485i·17-s − 1.74·21-s − 0.589i·23-s + 1.08i·27-s − 1.11·29-s − 1.01·31-s − 2.78i·33-s + 1.64i·37-s − 0.905·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.340807 - 1.44368i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.340807 - 1.44368i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.82iT - 3T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2.82iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.846679429463211994667375317806, −8.895197343666885667898512600132, −7.958069278784344041259958033565, −7.18908597827310180320708150664, −6.68303561172275608662172738468, −5.82006213838957153514853138375, −4.35669953428958015909749499938, −3.21253311380630296258376576171, −1.72420124670020385801849702489, −0.78057963022530357639902968446,
2.04723399175564830670440904490, 3.64844616579124356423310795620, 4.04150960796093900562391899915, 5.31302365798784626593101145035, 5.92072671130686155301539096266, 7.10343536019744415804637476350, 8.570608115548216271352625294723, 9.247653190571627922353877407335, 9.438407121700827425695138553788, 10.60177225633916216250728939269