| L(s) = 1 | − 0.786i·3-s − 2.08i·7-s + 2.38·9-s + 1.29·11-s − 1.21i·13-s + 4.08i·17-s + 19-s − 1.63·21-s + 8.95i·23-s − 4.23i·27-s + 9.38·29-s − 1.02i·33-s − 2i·37-s − 0.954·39-s + 3.57·41-s + ⋯ |
| L(s) = 1 | − 0.454i·3-s − 0.787i·7-s + 0.793·9-s + 0.391·11-s − 0.336i·13-s + 0.990i·17-s + 0.229·19-s − 0.357·21-s + 1.86i·23-s − 0.814i·27-s + 1.74·29-s − 0.177i·33-s − 0.328i·37-s − 0.152·39-s + 0.558·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.208014910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208014910\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.786iT - 3T^{2} \) |
| 7 | \( 1 + 2.08iT - 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 + 1.21iT - 13T^{2} \) |
| 17 | \( 1 - 4.08iT - 17T^{2} \) |
| 23 | \( 1 - 8.95iT - 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 + 7.72iT - 43T^{2} \) |
| 47 | \( 1 - 9.46iT - 47T^{2} \) |
| 53 | \( 1 - 11.9iT - 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 - 11.3iT - 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 + 5.65iT - 73T^{2} \) |
| 79 | \( 1 + 9.57T + 79T^{2} \) |
| 83 | \( 1 + 10.7iT - 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 8.59iT - 97T^{2} \) |
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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
−8.359358978996834426948912760163, −7.39126292465483577564087881675, −7.28912623260240307443371042354, −6.25654451205843257710606816012, −5.61090900370663890151781576155, −4.43371693057417858843882396154, −3.94189198185279426399946004820, −2.93727563754224441665475772800, −1.64361203875561117139629037145, −0.958907072719369109721223486395,
0.876420074098641096445477902280, 2.20096637864972396959167303667, 2.98651311181765647836770982558, 4.10017520662649047483958188422, 4.72914277668505858496706884140, 5.39658808189692646976470639208, 6.59512088330501239084789334218, 6.78945406764239395562763070359, 7.950691918058362496913368710552, 8.670554858659869132898576959691
