| L(s) = 1 | − 3-s − 5-s − 2.44i·7-s + 9-s + 5.73i·11-s + 3.19i·13-s + 15-s − 1.96·17-s + (0.679 − 4.30i)19-s + 2.44i·21-s + 7.49i·23-s + 25-s − 27-s − 6.19i·29-s − 4.75·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.923i·7-s + 0.333·9-s + 1.72i·11-s + 0.884i·13-s + 0.258·15-s − 0.475·17-s + (0.155 − 0.987i)19-s + 0.533i·21-s + 1.56i·23-s + 0.200·25-s − 0.192·27-s − 1.15i·29-s − 0.854·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07048855971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07048855971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (-0.679 + 4.30i)T \) |
| good | 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 5.73iT - 11T^{2} \) |
| 13 | \( 1 - 3.19iT - 13T^{2} \) |
| 17 | \( 1 + 1.96T + 17T^{2} \) |
| 23 | \( 1 - 7.49iT - 23T^{2} \) |
| 29 | \( 1 + 6.19iT - 29T^{2} \) |
| 31 | \( 1 + 4.75T + 31T^{2} \) |
| 37 | \( 1 + 6.54iT - 37T^{2} \) |
| 41 | \( 1 - 8.05iT - 41T^{2} \) |
| 43 | \( 1 + 3.08iT - 43T^{2} \) |
| 47 | \( 1 - 7.96iT - 47T^{2} \) |
| 53 | \( 1 + 6.48iT - 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 6.82T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 1.35T + 73T^{2} \) |
| 79 | \( 1 - 0.478T + 79T^{2} \) |
| 83 | \( 1 + 12.5iT - 83T^{2} \) |
| 89 | \( 1 + 16.2iT - 89T^{2} \) |
| 97 | \( 1 - 4.92iT - 97T^{2} \) |
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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
−7.56426633651165679523048240513, −7.33522024680635992555853061235, −6.79054707594948786953362314409, −5.82782552568922055746837480250, −4.79902183664841736483460097982, −4.36531520225183205785661654286, −3.69737658160892942584531411043, −2.31605236704039080243454339507, −1.39679423655832636260540981456, −0.02410910032055080451505305947,
1.08698834340663967835870464548, 2.49408909687933442290900280833, 3.30284048854520016649987730554, 4.09497208820756026241600096907, 5.27796571031051964711950623910, 5.60750344583420008650012853881, 6.34743352413705754618480540934, 7.10663870756275887137249838809, 8.156908985831513143266572845382, 8.539806294160582406850444923996
