| L(s) = 1 | + 0.510·2-s + 3-s − 1.73·4-s + 2.52·5-s + 0.510·6-s + 1.21·7-s − 1.90·8-s + 9-s + 1.28·10-s − 0.502·11-s − 1.73·12-s + 6.57·13-s + 0.619·14-s + 2.52·15-s + 2.50·16-s − 5.32·17-s + 0.510·18-s + 2.19·19-s − 4.38·20-s + 1.21·21-s − 0.256·22-s − 23-s − 1.90·24-s + 1.35·25-s + 3.35·26-s + 27-s − 2.10·28-s + ⋯ |
| L(s) = 1 | + 0.361·2-s + 0.577·3-s − 0.869·4-s + 1.12·5-s + 0.208·6-s + 0.458·7-s − 0.675·8-s + 0.333·9-s + 0.407·10-s − 0.151·11-s − 0.502·12-s + 1.82·13-s + 0.165·14-s + 0.650·15-s + 0.625·16-s − 1.29·17-s + 0.120·18-s + 0.503·19-s − 0.980·20-s + 0.264·21-s − 0.0546·22-s − 0.208·23-s − 0.389·24-s + 0.270·25-s + 0.658·26-s + 0.192·27-s − 0.398·28-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
| \(L(1)\) |
\(\approx\) |
\(2.867237285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.867237285\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.510T + 2T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 + 0.502T + 11T^{2} \) |
| 13 | \( 1 - 6.57T + 13T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 31 | \( 1 - 9.31T + 31T^{2} \) |
| 37 | \( 1 + 1.80T + 37T^{2} \) |
| 41 | \( 1 - 0.987T + 41T^{2} \) |
| 43 | \( 1 - 7.62T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 - 0.291T + 71T^{2} \) |
| 73 | \( 1 - 7.58T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 + 1.76T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 7.03T + 97T^{2} \) |
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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
−9.082024951764974787614079603683, −8.544414506738678519953244426735, −7.86940845398676010616424704377, −6.48947885695734816662225176263, −5.99559010045361091658739420660, −5.05326553640989199662677107170, −4.25971398518869637948997723645, −3.38417411170186963379537433141, −2.26893194439695875541869058169, −1.15217972032485095235666457765,
1.15217972032485095235666457765, 2.26893194439695875541869058169, 3.38417411170186963379537433141, 4.25971398518869637948997723645, 5.05326553640989199662677107170, 5.99559010045361091658739420660, 6.48947885695734816662225176263, 7.86940845398676010616424704377, 8.544414506738678519953244426735, 9.082024951764974787614079603683
