| L(s) = 1 | + 0.258·2-s − 1.93·4-s − 3.47·5-s − 4.42·7-s − 1.01·8-s − 0.896·10-s − 0.126·11-s − 13-s − 1.14·14-s + 3.60·16-s − 0.346·17-s − 0.863·19-s + 6.71·20-s − 0.0326·22-s + 8.11·23-s + 7.06·25-s − 0.258·26-s + 8.55·28-s + 2.50·29-s − 6.02·31-s + 2.96·32-s − 0.0894·34-s + 15.3·35-s − 2.44·37-s − 0.223·38-s + 3.52·40-s − 4.73·41-s + ⋯ |
| L(s) = 1 | + 0.182·2-s − 0.966·4-s − 1.55·5-s − 1.67·7-s − 0.359·8-s − 0.283·10-s − 0.0381·11-s − 0.277·13-s − 0.305·14-s + 0.901·16-s − 0.0839·17-s − 0.198·19-s + 1.50·20-s − 0.00696·22-s + 1.69·23-s + 1.41·25-s − 0.0506·26-s + 1.61·28-s + 0.464·29-s − 1.08·31-s + 0.523·32-s − 0.0153·34-s + 2.59·35-s − 0.402·37-s − 0.0361·38-s + 0.557·40-s − 0.739·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4650226038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4650226038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.258T + 2T^{2} \) |
| 5 | \( 1 + 3.47T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 0.126T + 11T^{2} \) |
| 17 | \( 1 + 0.346T + 17T^{2} \) |
| 19 | \( 1 + 0.863T + 19T^{2} \) |
| 23 | \( 1 - 8.11T + 23T^{2} \) |
| 29 | \( 1 - 2.50T + 29T^{2} \) |
| 31 | \( 1 + 6.02T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 + 0.214T + 47T^{2} \) |
| 53 | \( 1 - 4.02T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 2.72T + 61T^{2} \) |
| 67 | \( 1 - 4.01T + 67T^{2} \) |
| 71 | \( 1 + 8.55T + 71T^{2} \) |
| 73 | \( 1 + 9.17T + 73T^{2} \) |
| 79 | \( 1 - 0.432T + 79T^{2} \) |
| 83 | \( 1 - 1.04T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 0.178T + 97T^{2} \) |
| 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
−9.831861241212206133359024683566, −8.984337111038277335733777115849, −8.435494639707346682632491884651, −7.31024502272124964612936858279, −6.72322747191912987075685130396, −5.46575899030308020829388956795, −4.50795951605451193164074661496, −3.59555870814120686640007428368, −3.09089365669763004014493296999, −0.49120520387618413993859533583,
0.49120520387618413993859533583, 3.09089365669763004014493296999, 3.59555870814120686640007428368, 4.50795951605451193164074661496, 5.46575899030308020829388956795, 6.72322747191912987075685130396, 7.31024502272124964612936858279, 8.435494639707346682632491884651, 8.984337111038277335733777115849, 9.831861241212206133359024683566
