L(s) = 1 | + 0.765·3-s − 3.22·5-s − 4.55·7-s − 2.41·9-s − 3.45·11-s − 1.86·13-s − 2.46·15-s + 17-s − 6.05·19-s − 3.48·21-s + 3.04·23-s + 5.39·25-s − 4.14·27-s − 4.74·29-s + 2.29·31-s − 2.64·33-s + 14.6·35-s + 0.243·37-s − 1.42·39-s − 7.57·41-s − 11.0·43-s + 7.78·45-s − 0.420·47-s + 13.7·49-s + 0.765·51-s − 11.6·53-s + 11.1·55-s + ⋯ |
L(s) = 1 | + 0.441·3-s − 1.44·5-s − 1.72·7-s − 0.804·9-s − 1.04·11-s − 0.517·13-s − 0.637·15-s + 0.242·17-s − 1.38·19-s − 0.761·21-s + 0.634·23-s + 1.07·25-s − 0.797·27-s − 0.881·29-s + 0.412·31-s − 0.459·33-s + 2.48·35-s + 0.0399·37-s − 0.228·39-s − 1.18·41-s − 1.68·43-s + 1.16·45-s − 0.0613·47-s + 1.96·49-s + 0.107·51-s − 1.60·53-s + 1.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09872779870\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09872779870\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 0.765T + 3T^{2} \) |
| 5 | \( 1 + 3.22T + 5T^{2} \) |
| 7 | \( 1 + 4.55T + 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 23 | \( 1 - 3.04T + 23T^{2} \) |
| 29 | \( 1 + 4.74T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 - 0.243T + 37T^{2} \) |
| 41 | \( 1 + 7.57T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.420T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 4.15T + 67T^{2} \) |
| 71 | \( 1 - 7.72T + 71T^{2} \) |
| 73 | \( 1 - 8.51T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 1.34T + 83T^{2} \) |
| 89 | \( 1 - 4.23T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−8.174942237099686629410974159551, −7.998318880082889330952633722717, −6.96223226694674243516759858205, −6.45976795091417925423887690343, −5.44030756751384007840557480437, −4.56653014956626599578873245841, −3.46717706308028520699670753184, −3.24109800435173310989850069395, −2.30956462548974233192210901900, −0.15971283590652941051328433946,
0.15971283590652941051328433946, 2.30956462548974233192210901900, 3.24109800435173310989850069395, 3.46717706308028520699670753184, 4.56653014956626599578873245841, 5.44030756751384007840557480437, 6.45976795091417925423887690343, 6.96223226694674243516759858205, 7.998318880082889330952633722717, 8.174942237099686629410974159551