| L(s) = 1 | + 2-s − 3-s + 4-s − 2.26·5-s − 6-s + 3.32·7-s + 8-s + 9-s − 2.26·10-s − 5.42·11-s − 12-s + 5.07·13-s + 3.32·14-s + 2.26·15-s + 16-s − 0.475·17-s + 18-s + 3.32·19-s − 2.26·20-s − 3.32·21-s − 5.42·22-s + 23-s − 24-s + 0.114·25-s + 5.07·26-s − 27-s + 3.32·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.01·5-s − 0.408·6-s + 1.25·7-s + 0.353·8-s + 0.333·9-s − 0.715·10-s − 1.63·11-s − 0.288·12-s + 1.40·13-s + 0.888·14-s + 0.583·15-s + 0.250·16-s − 0.115·17-s + 0.235·18-s + 0.762·19-s − 0.505·20-s − 0.725·21-s − 1.15·22-s + 0.208·23-s − 0.204·24-s + 0.0228·25-s + 0.995·26-s − 0.192·27-s + 0.628·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.267206001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.267206001\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 + 5.42T + 11T^{2} \) |
| 13 | \( 1 - 5.07T + 13T^{2} \) |
| 17 | \( 1 + 0.475T + 17T^{2} \) |
| 19 | \( 1 - 3.32T + 19T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 7.67T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 8.38T + 53T^{2} \) |
| 59 | \( 1 - 8.47T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 6.27T + 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 + 4.26T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 5.80T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.112308176657256788733941374281, −7.68873172529163406885175849598, −7.13900320986407489093206406327, −5.85533943103063943820499118695, −5.50797416598101728531616926386, −4.64716879010523409752934287393, −4.05729770970461052444178969489, −3.15841805328741432209865876059, −2.01939445412549633272278083207, −0.812976330931551710405763141763,
0.812976330931551710405763141763, 2.01939445412549633272278083207, 3.15841805328741432209865876059, 4.05729770970461052444178969489, 4.64716879010523409752934287393, 5.50797416598101728531616926386, 5.85533943103063943820499118695, 7.13900320986407489093206406327, 7.68873172529163406885175849598, 8.112308176657256788733941374281
