| L(s) = 1 | − 2-s + 4-s − 2.78·5-s − 2.21·7-s − 8-s + 2.78·10-s − 11-s + 0.977·13-s + 2.21·14-s + 16-s − 7.92·17-s + 19-s − 2.78·20-s + 22-s − 1.02·23-s + 2.76·25-s − 0.977·26-s − 2.21·28-s − 3.80·29-s − 4.49·31-s − 32-s + 7.92·34-s + 6.16·35-s − 2.95·37-s − 38-s + 2.78·40-s + 3.31·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.24·5-s − 0.835·7-s − 0.353·8-s + 0.881·10-s − 0.301·11-s + 0.271·13-s + 0.590·14-s + 0.250·16-s − 1.92·17-s + 0.229·19-s − 0.623·20-s + 0.213·22-s − 0.213·23-s + 0.552·25-s − 0.191·26-s − 0.417·28-s − 0.707·29-s − 0.808·31-s − 0.176·32-s + 1.35·34-s + 1.04·35-s − 0.485·37-s − 0.162·38-s + 0.440·40-s + 0.517·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3322385714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3322385714\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.78T + 5T^{2} \) |
| 7 | \( 1 + 2.21T + 7T^{2} \) |
| 13 | \( 1 - 0.977T + 13T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 23 | \( 1 + 1.02T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 + 4.49T + 31T^{2} \) |
| 37 | \( 1 + 2.95T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 - 0.236T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 - 7.52T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 1.13T + 61T^{2} \) |
| 67 | \( 1 - 4.35T + 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 - 1.53T + 73T^{2} \) |
| 79 | \( 1 + 7.01T + 79T^{2} \) |
| 83 | \( 1 - 3.31T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−8.595106246587903644348277897280, −7.76510341662592968695141054910, −7.17213692253874559001951615103, −6.54732978509392928095530686560, −5.69365001247478908566129585552, −4.52587678529126126195174397741, −3.79758214121094209836604188574, −2.99342368764984659903806855563, −1.93089431857362079195219434723, −0.35233284420115496940433269703,
0.35233284420115496940433269703, 1.93089431857362079195219434723, 2.99342368764984659903806855563, 3.79758214121094209836604188574, 4.52587678529126126195174397741, 5.69365001247478908566129585552, 6.54732978509392928095530686560, 7.17213692253874559001951615103, 7.76510341662592968695141054910, 8.595106246587903644348277897280
