| L(s) = 1 | − 2-s − 2.26·3-s + 4-s − 5-s + 2.26·6-s − 4.08·7-s − 8-s + 2.14·9-s + 10-s + 11-s − 2.26·12-s − 1.93·13-s + 4.08·14-s + 2.26·15-s + 16-s − 0.878·17-s − 2.14·18-s + 1.79·19-s − 20-s + 9.25·21-s − 22-s − 7.27·23-s + 2.26·24-s + 25-s + 1.93·26-s + 1.94·27-s − 4.08·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.30·3-s + 0.5·4-s − 0.447·5-s + 0.925·6-s − 1.54·7-s − 0.353·8-s + 0.714·9-s + 0.316·10-s + 0.301·11-s − 0.654·12-s − 0.536·13-s + 1.09·14-s + 0.585·15-s + 0.250·16-s − 0.212·17-s − 0.505·18-s + 0.412·19-s − 0.223·20-s + 2.02·21-s − 0.213·22-s − 1.51·23-s + 0.462·24-s + 0.200·25-s + 0.379·26-s + 0.373·27-s − 0.771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03097642272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03097642272\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 + 0.878T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 + 2.05T + 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 + 8.41T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 47 | \( 1 + 4.81T + 47T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 9.69T + 67T^{2} \) |
| 71 | \( 1 - 5.44T + 71T^{2} \) |
| 73 | \( 1 - 9.20T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 4.42T + 83T^{2} \) |
| 89 | \( 1 - 6.02T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.286109927766222935946513736558, −7.44037121567911430327063728489, −6.65362080303101519616865433941, −6.37527597698016268678181540044, −5.57924222927700913362884674153, −4.72415025321025721655282693831, −3.66981923515337215552452691874, −2.92588309796639875644649798693, −1.57706983608932566473228365579, −0.11442449909268821443411722168,
0.11442449909268821443411722168, 1.57706983608932566473228365579, 2.92588309796639875644649798693, 3.66981923515337215552452691874, 4.72415025321025721655282693831, 5.57924222927700913362884674153, 6.37527597698016268678181540044, 6.65362080303101519616865433941, 7.44037121567911430327063728489, 8.286109927766222935946513736558
