| L(s) = 1 | − 1.41·2-s + 3-s + 0.00250·4-s + 0.0661·5-s − 1.41·6-s + 1.68·7-s + 2.82·8-s + 9-s − 0.0936·10-s − 6.36·11-s + 0.00250·12-s + 1.48·13-s − 2.38·14-s + 0.0661·15-s − 4.00·16-s − 0.164·17-s − 1.41·18-s + 4.96·19-s + 0.000165·20-s + 1.68·21-s + 9.00·22-s + 23-s + 2.82·24-s − 4.99·25-s − 2.09·26-s + 27-s + 0.00421·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.577·3-s + 0.00125·4-s + 0.0295·5-s − 0.577·6-s + 0.636·7-s + 0.999·8-s + 0.333·9-s − 0.0296·10-s − 1.91·11-s + 0.000722·12-s + 0.411·13-s − 0.636·14-s + 0.0170·15-s − 1.00·16-s − 0.0400·17-s − 0.333·18-s + 1.13·19-s + 3.70e−5·20-s + 0.367·21-s + 1.91·22-s + 0.208·23-s + 0.576·24-s − 0.999·25-s − 0.411·26-s + 0.192·27-s + 0.000795·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143693235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143693235\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 - 0.0661T + 5T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 + 6.36T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 + 0.164T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 0.107T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 2.20T + 79T^{2} \) |
| 83 | \( 1 - 8.12T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−9.250100845303621855652174881399, −8.184981805871715120482905347910, −7.85538373072156271927436857552, −7.42039484254129127800016132742, −5.98794449838483864359021035802, −5.04278499259284416564602157829, −4.32835536980815219644654947608, −3.01959671856565568670779909902, −2.07010984252047416838597490245, −0.822193626087309992395528299991,
0.822193626087309992395528299991, 2.07010984252047416838597490245, 3.01959671856565568670779909902, 4.32835536980815219644654947608, 5.04278499259284416564602157829, 5.98794449838483864359021035802, 7.42039484254129127800016132742, 7.85538373072156271927436857552, 8.184981805871715120482905347910, 9.250100845303621855652174881399
