| L(s) = 1 | − 3.24·2-s + 3·3-s + 2.52·4-s − 19.5·5-s − 9.73·6-s + 36.4·7-s + 17.7·8-s + 9·9-s + 63.5·10-s − 30.4·11-s + 7.57·12-s − 16.3·13-s − 118.·14-s − 58.7·15-s − 77.8·16-s + 57.0·17-s − 29.1·18-s + 127.·19-s − 49.4·20-s + 109.·21-s + 98.6·22-s + 23·23-s + 53.2·24-s + 258.·25-s + 53.0·26-s + 27·27-s + 91.9·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.315·4-s − 1.75·5-s − 0.662·6-s + 1.96·7-s + 0.785·8-s + 0.333·9-s + 2.00·10-s − 0.833·11-s + 0.182·12-s − 0.348·13-s − 2.25·14-s − 1.01·15-s − 1.21·16-s + 0.813·17-s − 0.382·18-s + 1.54·19-s − 0.552·20-s + 1.13·21-s + 0.955·22-s + 0.208·23-s + 0.453·24-s + 2.07·25-s + 0.399·26-s + 0.192·27-s + 0.620·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.242063194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242063194\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 3.24T + 8T^{2} \) |
| 5 | \( 1 + 19.5T + 125T^{2} \) |
| 7 | \( 1 - 36.4T + 343T^{2} \) |
| 11 | \( 1 + 30.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 31 | \( 1 + 17.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 50.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 156.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 98.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 603.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 664.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 295.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 168.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 225.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 475.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 739.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 616.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 269.T + 9.12e5T^{2} \) |
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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
−8.581551969484195926764926933404, −7.895722372813461549093235081616, −7.65881823003180347953888048556, −7.31991825202460587347663994366, −5.27938044835086221900961045703, −4.69772938114810998257960599868, −3.90032934498418360552652647152, −2.74689235409755672048448046734, −1.45370063412742272967300482841, −0.66054869361478970405567054804,
0.66054869361478970405567054804, 1.45370063412742272967300482841, 2.74689235409755672048448046734, 3.90032934498418360552652647152, 4.69772938114810998257960599868, 5.27938044835086221900961045703, 7.31991825202460587347663994366, 7.65881823003180347953888048556, 7.895722372813461549093235081616, 8.581551969484195926764926933404
