| L(s) = 1 | + 2.36·2-s − 2.87·3-s + 3.58·4-s − 6.79·6-s + 0.269·7-s + 3.74·8-s + 5.25·9-s − 11-s − 10.3·12-s + 1.50·13-s + 0.636·14-s + 1.68·16-s − 3.65·17-s + 12.4·18-s − 19-s − 0.774·21-s − 2.36·22-s − 2.55·23-s − 10.7·24-s + 3.56·26-s − 6.48·27-s + 0.965·28-s + 0.572·29-s + 5.82·31-s − 3.51·32-s + 2.87·33-s − 8.64·34-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 1.65·3-s + 1.79·4-s − 2.77·6-s + 0.101·7-s + 1.32·8-s + 1.75·9-s − 0.301·11-s − 2.97·12-s + 0.418·13-s + 0.170·14-s + 0.420·16-s − 0.887·17-s + 2.92·18-s − 0.229·19-s − 0.168·21-s − 0.503·22-s − 0.533·23-s − 2.19·24-s + 0.699·26-s − 1.24·27-s + 0.182·28-s + 0.106·29-s + 1.04·31-s − 0.621·32-s + 0.500·33-s − 1.48·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 7 | \( 1 - 0.269T + 7T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 - 0.572T + 29T^{2} \) |
| 31 | \( 1 - 5.82T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 3.61T + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 - 3.41T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 0.482T + 73T^{2} \) |
| 79 | \( 1 + 1.93T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 - 0.150T + 89T^{2} \) |
| 97 | \( 1 - 1.43T + 97T^{2} \) |
| 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
−7.36401220046601225092260355235, −6.63567226114071228355520294794, −6.15020969539126534897779697055, −5.68704348239658187304288251233, −4.76307279181875486703331439382, −4.56265659441558939684051982689, −3.63781285720558863062882860005, −2.57947618940417147967218347496, −1.48857561925687474626953134017, 0,
1.48857561925687474626953134017, 2.57947618940417147967218347496, 3.63781285720558863062882860005, 4.56265659441558939684051982689, 4.76307279181875486703331439382, 5.68704348239658187304288251233, 6.15020969539126534897779697055, 6.63567226114071228355520294794, 7.36401220046601225092260355235
