| L(s) = 1 | − 2.69·2-s − 2.81·3-s + 5.27·4-s − 5-s + 7.59·6-s − 1.69·7-s − 8.83·8-s + 4.92·9-s + 2.69·10-s + 3.99·11-s − 14.8·12-s + 0.417·13-s + 4.57·14-s + 2.81·15-s + 13.2·16-s + 3.25·17-s − 13.2·18-s − 5.00·19-s − 5.27·20-s + 4.77·21-s − 10.7·22-s + 9.05·23-s + 24.8·24-s + 25-s − 1.12·26-s − 5.41·27-s − 8.93·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 1.62·3-s + 2.63·4-s − 0.447·5-s + 3.09·6-s − 0.640·7-s − 3.12·8-s + 1.64·9-s + 0.852·10-s + 1.20·11-s − 4.28·12-s + 0.115·13-s + 1.22·14-s + 0.726·15-s + 3.31·16-s + 0.789·17-s − 3.13·18-s − 1.14·19-s − 1.17·20-s + 1.04·21-s − 2.29·22-s + 1.88·23-s + 5.07·24-s + 0.200·25-s − 0.220·26-s − 1.04·27-s − 1.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3181442501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3181442501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 + 2.81T + 3T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 - 3.99T + 11T^{2} \) |
| 13 | \( 1 - 0.417T + 13T^{2} \) |
| 17 | \( 1 - 3.25T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 23 | \( 1 - 9.05T + 23T^{2} \) |
| 29 | \( 1 - 0.120T + 29T^{2} \) |
| 31 | \( 1 - 1.05T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 + 0.542T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 5.31T + 47T^{2} \) |
| 53 | \( 1 + 5.09T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 7.86T + 79T^{2} \) |
| 83 | \( 1 - 1.88T + 83T^{2} \) |
| 89 | \( 1 + 7.72T + 89T^{2} \) |
| 97 | \( 1 + 4.83T + 97T^{2} \) |
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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
−9.121744911784486200899281638879, −8.646354338893076139383471986466, −7.42225661661247406832873205955, −6.92306599553396906118903654078, −6.32188225723683326106054176481, −5.64029447559672488106733007952, −4.27035995110282183698170988082, −2.98746547064170956986562521152, −1.40309581640502101910303884561, −0.59671903263855743750289931470,
0.59671903263855743750289931470, 1.40309581640502101910303884561, 2.98746547064170956986562521152, 4.27035995110282183698170988082, 5.64029447559672488106733007952, 6.32188225723683326106054176481, 6.92306599553396906118903654078, 7.42225661661247406832873205955, 8.646354338893076139383471986466, 9.121744911784486200899281638879
