| L(s) = 1 | − 2.17·2-s + 3.11·3-s + 2.71·4-s − 5-s − 6.76·6-s − 7-s − 1.55·8-s + 6.69·9-s + 2.17·10-s − 0.102·11-s + 8.46·12-s + 3.05·13-s + 2.17·14-s − 3.11·15-s − 2.04·16-s + 6.79·17-s − 14.5·18-s + 7.10·19-s − 2.71·20-s − 3.11·21-s + 0.223·22-s − 6.34·23-s − 4.85·24-s + 25-s − 6.63·26-s + 11.4·27-s − 2.71·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.79·3-s + 1.35·4-s − 0.447·5-s − 2.76·6-s − 0.377·7-s − 0.551·8-s + 2.23·9-s + 0.686·10-s − 0.0310·11-s + 2.44·12-s + 0.847·13-s + 0.580·14-s − 0.803·15-s − 0.512·16-s + 1.64·17-s − 3.42·18-s + 1.63·19-s − 0.607·20-s − 0.679·21-s + 0.0476·22-s − 1.32·23-s − 0.991·24-s + 0.200·25-s − 1.30·26-s + 2.21·27-s − 0.513·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.012787288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.012787288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 - 3.11T + 3T^{2} \) |
| 11 | \( 1 + 0.102T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 - 7.10T + 19T^{2} \) |
| 23 | \( 1 + 6.34T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 6.66T + 31T^{2} \) |
| 37 | \( 1 - 3.36T + 37T^{2} \) |
| 41 | \( 1 - 0.870T + 41T^{2} \) |
| 43 | \( 1 + 6.76T + 43T^{2} \) |
| 47 | \( 1 + 2.90T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + 5.23T + 59T^{2} \) |
| 61 | \( 1 - 8.37T + 61T^{2} \) |
| 67 | \( 1 - 7.08T + 67T^{2} \) |
| 71 | \( 1 + 7.78T + 71T^{2} \) |
| 73 | \( 1 - 7.59T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 2.05T + 89T^{2} \) |
| 97 | \( 1 - 1.16T + 97T^{2} \) |
| 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
−8.081665846045997804926134953615, −7.56592194895246098912396065572, −7.01961291743415918332242148769, −6.07465077837851020470735997476, −4.94082884665821545946142891196, −3.62805712790673815124263209269, −3.52445674380304240439709400748, −2.52522598462683779559401023253, −1.61363784409008903350025879844, −0.872673155224118248932714049634,
0.872673155224118248932714049634, 1.61363784409008903350025879844, 2.52522598462683779559401023253, 3.52445674380304240439709400748, 3.62805712790673815124263209269, 4.94082884665821545946142891196, 6.07465077837851020470735997476, 7.01961291743415918332242148769, 7.56592194895246098912396065572, 8.081665846045997804926134953615
