| L(s) = 1 | + 2-s + 3-s + 4-s − 2.28·5-s + 6-s + 7-s + 8-s + 9-s − 2.28·10-s + 11-s + 12-s + 13-s + 14-s − 2.28·15-s + 16-s − 7.05·17-s + 18-s + 7.81·19-s − 2.28·20-s + 21-s + 22-s − 6.18·23-s + 24-s + 0.204·25-s + 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.02·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.721·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.589·15-s + 0.250·16-s − 1.71·17-s + 0.235·18-s + 1.79·19-s − 0.510·20-s + 0.218·21-s + 0.213·22-s − 1.28·23-s + 0.204·24-s + 0.0408·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.572425270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.572425270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.28T + 5T^{2} \) |
| 17 | \( 1 + 7.05T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 23 | \( 1 + 6.18T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 1.76T + 41T^{2} \) |
| 43 | \( 1 + 3.57T + 43T^{2} \) |
| 47 | \( 1 - 1.16T + 47T^{2} \) |
| 53 | \( 1 + 5.38T + 53T^{2} \) |
| 59 | \( 1 + 3.15T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 - 4.88T + 73T^{2} \) |
| 79 | \( 1 - 0.310T + 79T^{2} \) |
| 83 | \( 1 - 9.59T + 83T^{2} \) |
| 89 | \( 1 + 4.16T + 89T^{2} \) |
| 97 | \( 1 - 0.544T + 97T^{2} \) |
| 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
−8.070243361152791670526240800326, −7.44946692534606791218633950215, −6.63566948912338870039960336256, −6.04309444055925939829850623602, −4.85968177358465927428108174649, −4.40163262257246390602777290799, −3.73969323591735564406739460511, −2.94427752823675412523622239856, −2.10015743262251292004044238443, −0.883254452082244791630862586176,
0.883254452082244791630862586176, 2.10015743262251292004044238443, 2.94427752823675412523622239856, 3.73969323591735564406739460511, 4.40163262257246390602777290799, 4.85968177358465927428108174649, 6.04309444055925939829850623602, 6.63566948912338870039960336256, 7.44946692534606791218633950215, 8.070243361152791670526240800326
