| L(s) = 1 | + 1.73·7-s − 3.46·19-s − 5·25-s + 1.73·31-s − 6.92·37-s − 5·43-s − 4·49-s + 61-s − 15.5·67-s + 15.5·73-s − 17·79-s + 5.19·97-s + 7·103-s − 12.1·109-s + ⋯ |
| L(s) = 1 | + 0.654·7-s − 0.794·19-s − 25-s + 0.311·31-s − 1.13·37-s − 0.762·43-s − 0.571·49-s + 0.128·61-s − 1.90·67-s + 1.82·73-s − 1.91·79-s + 0.527·97-s + 0.689·103-s − 1.16·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 17T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81766255114161347993477749730, −7.01882926698614787657487486010, −6.31234789372820725681681334745, −5.54638297763503232778644905724, −4.80321683734389458223072062275, −4.11056584032868454838640183450, −3.25188958921145096612546032330, −2.20513435705524486513020177223, −1.44541016615256329290115722772, 0,
1.44541016615256329290115722772, 2.20513435705524486513020177223, 3.25188958921145096612546032330, 4.11056584032868454838640183450, 4.80321683734389458223072062275, 5.54638297763503232778644905724, 6.31234789372820725681681334745, 7.01882926698614787657487486010, 7.81766255114161347993477749730
