L(s) = 1 | − 2-s − 0.876·3-s + 4-s − 5-s + 0.876·6-s + 4.75·7-s − 8-s − 2.23·9-s + 10-s + 0.395·11-s − 0.876·12-s − 1.38·13-s − 4.75·14-s + 0.876·15-s + 16-s + 1.97·17-s + 2.23·18-s − 4.95·19-s − 20-s − 4.16·21-s − 0.395·22-s + 7.99·23-s + 0.876·24-s + 25-s + 1.38·26-s + 4.58·27-s + 4.75·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.505·3-s + 0.5·4-s − 0.447·5-s + 0.357·6-s + 1.79·7-s − 0.353·8-s − 0.744·9-s + 0.316·10-s + 0.119·11-s − 0.252·12-s − 0.385·13-s − 1.27·14-s + 0.226·15-s + 0.250·16-s + 0.479·17-s + 0.526·18-s − 1.13·19-s − 0.223·20-s − 0.909·21-s − 0.0843·22-s + 1.66·23-s + 0.178·24-s + 0.200·25-s + 0.272·26-s + 0.882·27-s + 0.899·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 0.876T + 3T^{2} \) |
| 7 | \( 1 - 4.75T + 7T^{2} \) |
| 11 | \( 1 - 0.395T + 11T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 17 | \( 1 - 1.97T + 17T^{2} \) |
| 19 | \( 1 + 4.95T + 19T^{2} \) |
| 23 | \( 1 - 7.99T + 23T^{2} \) |
| 29 | \( 1 + 5.94T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 + 7.50T + 37T^{2} \) |
| 41 | \( 1 + 3.64T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 9.91T + 53T^{2} \) |
| 59 | \( 1 - 9.82T + 59T^{2} \) |
| 61 | \( 1 + 5.61T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 - 9.52T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 7.21T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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
−8.175837552201961683712883923696, −7.41670184254555900190193349897, −6.89914210073620948550113882631, −5.70629468454502811707229460422, −5.21283191101860258451123468265, −4.43699695566273733044013454409, −3.32909929477339773725148402407, −2.18097414017264382917163612671, −1.30616904632550181407291907193, 0,
1.30616904632550181407291907193, 2.18097414017264382917163612671, 3.32909929477339773725148402407, 4.43699695566273733044013454409, 5.21283191101860258451123468265, 5.70629468454502811707229460422, 6.89914210073620948550113882631, 7.41670184254555900190193349897, 8.175837552201961683712883923696