L(s) = 1 | + 2-s − 3-s + 4-s + 2.13·5-s − 6-s − 0.0489·7-s + 8-s + 9-s + 2.13·10-s + 6.29·11-s − 12-s − 0.0489·14-s − 2.13·15-s + 16-s − 2.89·17-s + 18-s − 7.20·19-s + 2.13·20-s + 0.0489·21-s + 6.29·22-s + 2.71·23-s − 24-s − 0.432·25-s − 27-s − 0.0489·28-s + 4.91·29-s − 2.13·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.955·5-s − 0.408·6-s − 0.0184·7-s + 0.353·8-s + 0.333·9-s + 0.675·10-s + 1.89·11-s − 0.288·12-s − 0.0130·14-s − 0.551·15-s + 0.250·16-s − 0.700·17-s + 0.235·18-s − 1.65·19-s + 0.477·20-s + 0.0106·21-s + 1.34·22-s + 0.565·23-s − 0.204·24-s − 0.0865·25-s − 0.192·27-s − 0.00924·28-s + 0.912·29-s − 0.390·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.628514863\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.628514863\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2.13T + 5T^{2} \) |
| 7 | \( 1 + 0.0489T + 7T^{2} \) |
| 11 | \( 1 - 6.29T + 11T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 + 7.20T + 19T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 - 4.91T + 29T^{2} \) |
| 31 | \( 1 - 9.00T + 31T^{2} \) |
| 37 | \( 1 + 0.176T + 37T^{2} \) |
| 41 | \( 1 + 8.59T + 41T^{2} \) |
| 43 | \( 1 - 6.71T + 43T^{2} \) |
| 47 | \( 1 - 7.20T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 + 4.26T + 59T^{2} \) |
| 61 | \( 1 - 7.10T + 61T^{2} \) |
| 67 | \( 1 - 5.38T + 67T^{2} \) |
| 71 | \( 1 + 8.71T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 + 2.47T + 97T^{2} \) |
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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
−10.11876693612205881537779626935, −9.199364012043754611330712347735, −8.426917069270007905553237765455, −6.81378051354124590363795940283, −6.52324786766301180099130336963, −5.79597003784574618625920826059, −4.63013126634936077953335818758, −3.99184081576424569928344067595, −2.47826276801074121661477756091, −1.34806012139143938612331236626,
1.34806012139143938612331236626, 2.47826276801074121661477756091, 3.99184081576424569928344067595, 4.63013126634936077953335818758, 5.79597003784574618625920826059, 6.52324786766301180099130336963, 6.81378051354124590363795940283, 8.426917069270007905553237765455, 9.199364012043754611330712347735, 10.11876693612205881537779626935