| L(s) = 1 | − 2-s − 3.07·3-s + 4-s + 0.0807·5-s + 3.07·6-s − 2.46·7-s − 8-s + 6.43·9-s − 0.0807·10-s − 4.79·11-s − 3.07·12-s + 0.832·13-s + 2.46·14-s − 0.248·15-s + 16-s + 3.88·17-s − 6.43·18-s − 3.52·19-s + 0.0807·20-s + 7.58·21-s + 4.79·22-s − 3.78·23-s + 3.07·24-s − 4.99·25-s − 0.832·26-s − 10.5·27-s − 2.46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.77·3-s + 0.5·4-s + 0.0361·5-s + 1.25·6-s − 0.933·7-s − 0.353·8-s + 2.14·9-s − 0.0255·10-s − 1.44·11-s − 0.886·12-s + 0.230·13-s + 0.659·14-s − 0.0640·15-s + 0.250·16-s + 0.943·17-s − 1.51·18-s − 0.809·19-s + 0.0180·20-s + 1.65·21-s + 1.02·22-s − 0.789·23-s + 0.626·24-s − 0.998·25-s − 0.163·26-s − 2.03·27-s − 0.466·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1668975890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1668975890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 - 0.0807T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 4.79T + 11T^{2} \) |
| 13 | \( 1 - 0.832T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 + 3.78T + 23T^{2} \) |
| 29 | \( 1 - 2.91T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 - 4.00T + 37T^{2} \) |
| 41 | \( 1 - 3.13T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 7.08T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 - 1.67T + 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 - 0.958T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 5.63T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.218412950180617833724315985797, −7.73066530301644746044171908117, −6.82621525008973997670293198261, −6.18558348898981121066374271332, −5.72405915566054241609315430225, −4.97076805264398904043769848289, −3.94986041075494146539980290688, −2.82049544253964160797952791493, −1.58821243096266903126083916553, −0.28397829254299255439331391900,
0.28397829254299255439331391900, 1.58821243096266903126083916553, 2.82049544253964160797952791493, 3.94986041075494146539980290688, 4.97076805264398904043769848289, 5.72405915566054241609315430225, 6.18558348898981121066374271332, 6.82621525008973997670293198261, 7.73066530301644746044171908117, 8.218412950180617833724315985797
