L(s) = 1 | − 2-s + 2.12·3-s + 4-s − 3.12·5-s − 2.12·6-s + 0.515·7-s − 8-s + 1.51·9-s + 3.12·10-s + 2.12·12-s + 0.484·13-s − 0.515·14-s − 6.64·15-s + 16-s + 1.51·17-s − 1.51·18-s + 19-s − 3.12·20-s + 1.09·21-s + 0.515·23-s − 2.12·24-s + 4.76·25-s − 0.484·26-s − 3.15·27-s + 0.515·28-s + 2.60·29-s + 6.64·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.22·3-s + 0.5·4-s − 1.39·5-s − 0.867·6-s + 0.194·7-s − 0.353·8-s + 0.505·9-s + 0.988·10-s + 0.613·12-s + 0.134·13-s − 0.137·14-s − 1.71·15-s + 0.250·16-s + 0.367·17-s − 0.357·18-s + 0.229·19-s − 0.698·20-s + 0.238·21-s + 0.107·23-s − 0.433·24-s + 0.952·25-s − 0.0950·26-s − 0.607·27-s + 0.0973·28-s + 0.484·29-s + 1.21·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.12T + 3T^{2} \) |
| 5 | \( 1 + 3.12T + 5T^{2} \) |
| 7 | \( 1 - 0.515T + 7T^{2} \) |
| 13 | \( 1 - 0.484T + 13T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 23 | \( 1 - 0.515T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 - 9.01T + 47T^{2} \) |
| 53 | \( 1 + 0.670T + 53T^{2} \) |
| 59 | \( 1 + 2.90T + 59T^{2} \) |
| 61 | \( 1 + 9.92T + 61T^{2} \) |
| 67 | \( 1 + 2.06T + 67T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 - 7.52T + 73T^{2} \) |
| 79 | \( 1 + 1.60T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 + 1.06T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
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
−7.999044085694893499534019578090, −7.54003604934483863362849405568, −6.99771705989168424709294436330, −5.87501129346001379453575560776, −4.81399837727392629823229141288, −3.81129055428732513336465706282, −3.35440507180873319332944901887, −2.48417433434070318071782463436, −1.38283292412009123673421104212, 0,
1.38283292412009123673421104212, 2.48417433434070318071782463436, 3.35440507180873319332944901887, 3.81129055428732513336465706282, 4.81399837727392629823229141288, 5.87501129346001379453575560776, 6.99771705989168424709294436330, 7.54003604934483863362849405568, 7.999044085694893499534019578090