L(s) = 1 | − 5-s + 2·11-s − 6·17-s − 8·19-s − 23-s + 25-s + 8·29-s + 8·31-s − 2·37-s − 6·41-s − 6·43-s − 4·47-s − 7·49-s + 6·53-s − 2·55-s + 2·61-s + 2·67-s + 14·71-s + 6·73-s − 16·79-s + 6·85-s − 16·89-s + 8·95-s − 4·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.603·11-s − 1.45·17-s − 1.83·19-s − 0.208·23-s + 1/5·25-s + 1.48·29-s + 1.43·31-s − 0.328·37-s − 0.937·41-s − 0.914·43-s − 0.583·47-s − 49-s + 0.824·53-s − 0.269·55-s + 0.256·61-s + 0.244·67-s + 1.66·71-s + 0.702·73-s − 1.80·79-s + 0.650·85-s − 1.69·89-s + 0.820·95-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.298233028\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.298233028\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.63901967798773, −15.50806051597059, −14.93013837636325, −14.22515848431024, −13.77555558282512, −13.06821341301116, −12.64074291901604, −11.94841583442417, −11.46818824108025, −10.95646270576225, −10.23544258165474, −9.844659983362298, −8.873743813039707, −8.414045620860337, −8.206999212093591, −7.040668115684820, −6.528776389765051, −6.342482518324426, −5.155626430340493, −4.472595789135047, −4.144275179547222, −3.214592311534061, −2.404544150912695, −1.656402914763373, −0.4792294441504235,
0.4792294441504235, 1.656402914763373, 2.404544150912695, 3.214592311534061, 4.144275179547222, 4.472595789135047, 5.155626430340493, 6.342482518324426, 6.528776389765051, 7.040668115684820, 8.206999212093591, 8.414045620860337, 8.873743813039707, 9.844659983362298, 10.23544258165474, 10.95646270576225, 11.46818824108025, 11.94841583442417, 12.64074291901604, 13.06821341301116, 13.77555558282512, 14.22515848431024, 14.93013837636325, 15.50806051597059, 15.63901967798773