| L(s) = 1 | − 2-s + 0.292·3-s + 4-s − 2.36·5-s − 0.292·6-s + 3.96·7-s − 8-s − 2.91·9-s + 2.36·10-s − 4.62·11-s + 0.292·12-s − 3.16·13-s − 3.96·14-s − 0.692·15-s + 16-s + 0.0361·17-s + 2.91·18-s + 1.07·19-s − 2.36·20-s + 1.15·21-s + 4.62·22-s + 4.68·23-s − 0.292·24-s + 0.615·25-s + 3.16·26-s − 1.72·27-s + 3.96·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.168·3-s + 0.5·4-s − 1.05·5-s − 0.119·6-s + 1.49·7-s − 0.353·8-s − 0.971·9-s + 0.749·10-s − 1.39·11-s + 0.0843·12-s − 0.878·13-s − 1.05·14-s − 0.178·15-s + 0.250·16-s + 0.00877·17-s + 0.687·18-s + 0.247·19-s − 0.529·20-s + 0.252·21-s + 0.985·22-s + 0.975·23-s − 0.0596·24-s + 0.123·25-s + 0.621·26-s − 0.332·27-s + 0.749·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 - 0.292T + 3T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 11 | \( 1 + 4.62T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 - 0.0361T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 + 9.64T + 29T^{2} \) |
| 31 | \( 1 + 4.06T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 + 2.80T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 + 2.17T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 - 3.42T + 67T^{2} \) |
| 71 | \( 1 - 8.59T + 71T^{2} \) |
| 73 | \( 1 + 8.39T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
| 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
−10.67151927477544505759243361131, −9.285294297490040221701412859230, −8.460022227700454746247557283704, −7.65558222085611238815612500658, −7.43694117209567088183996382251, −5.55230044375907880073276032682, −4.83145098301425494316976901471, −3.30413003462160399027029545788, −2.06266475400692312260068191156, 0,
2.06266475400692312260068191156, 3.30413003462160399027029545788, 4.83145098301425494316976901471, 5.55230044375907880073276032682, 7.43694117209567088183996382251, 7.65558222085611238815612500658, 8.460022227700454746247557283704, 9.285294297490040221701412859230, 10.67151927477544505759243361131
