| L(s) = 1 | + 2-s + 3-s + 4-s + 3.95·5-s + 6-s + 7-s + 8-s + 9-s + 3.95·10-s + 3.91·11-s + 12-s + 3.15·13-s + 14-s + 3.95·15-s + 16-s − 0.274·17-s + 18-s − 5.15·19-s + 3.95·20-s + 21-s + 3.91·22-s − 2.79·23-s + 24-s + 10.6·25-s + 3.15·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.76·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.25·10-s + 1.18·11-s + 0.288·12-s + 0.875·13-s + 0.267·14-s + 1.02·15-s + 0.250·16-s − 0.0664·17-s + 0.235·18-s − 1.18·19-s + 0.884·20-s + 0.218·21-s + 0.835·22-s − 0.582·23-s + 0.204·24-s + 2.12·25-s + 0.618·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.016366982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.016366982\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 5 | \( 1 - 3.95T + 5T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 + 0.274T + 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 23 | \( 1 + 2.79T + 23T^{2} \) |
| 29 | \( 1 + 5.61T + 29T^{2} \) |
| 31 | \( 1 - 1.34T + 31T^{2} \) |
| 37 | \( 1 + 5.79T + 37T^{2} \) |
| 41 | \( 1 - 2.13T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 6.70T + 53T^{2} \) |
| 59 | \( 1 - 5.03T + 59T^{2} \) |
| 61 | \( 1 - 7.96T + 61T^{2} \) |
| 67 | \( 1 - 0.669T + 67T^{2} \) |
| 71 | \( 1 - 5.21T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 3.58T + 79T^{2} \) |
| 83 | \( 1 - 5.76T + 83T^{2} \) |
| 89 | \( 1 + 5.53T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
| 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
−7.82382168711156694514716257326, −6.80013234474947149990573427097, −6.36409312610227852231912035004, −5.84181680820636952152271238175, −5.08408415419123963504145149908, −4.20285347663347876861352665225, −3.60590431743520352561485169496, −2.55616524407693069599995031039, −1.85826469319236052969182407752, −1.34568818154375707398114122966,
1.34568818154375707398114122966, 1.85826469319236052969182407752, 2.55616524407693069599995031039, 3.60590431743520352561485169496, 4.20285347663347876861352665225, 5.08408415419123963504145149908, 5.84181680820636952152271238175, 6.36409312610227852231912035004, 6.80013234474947149990573427097, 7.82382168711156694514716257326
