L(s) = 1 | + 1.54·2-s + 0.373·4-s − 3.37·7-s − 2.50·8-s − 0.275·11-s − 5.74·13-s − 5.20·14-s − 4.60·16-s + 6.11·17-s − 5.04·19-s − 0.424·22-s + 2.66·23-s − 8.84·26-s − 1.26·28-s + 6.27·29-s + 0.987·31-s − 2.08·32-s + 9.42·34-s − 6.20·37-s − 7.77·38-s + 0.876·41-s − 0.0269·43-s − 0.103·44-s + 4.10·46-s + 11.2·47-s + 4.39·49-s − 2.14·52-s + ⋯ |
L(s) = 1 | + 1.08·2-s + 0.186·4-s − 1.27·7-s − 0.885·8-s − 0.0830·11-s − 1.59·13-s − 1.39·14-s − 1.15·16-s + 1.48·17-s − 1.15·19-s − 0.0904·22-s + 0.555·23-s − 1.73·26-s − 0.238·28-s + 1.16·29-s + 0.177·31-s − 0.369·32-s + 1.61·34-s − 1.02·37-s − 1.26·38-s + 0.136·41-s − 0.00411·43-s − 0.0155·44-s + 0.605·46-s + 1.64·47-s + 0.628·49-s − 0.297·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.788635623\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.788635623\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 0.275T + 11T^{2} \) |
| 13 | \( 1 + 5.74T + 13T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 31 | \( 1 - 0.987T + 31T^{2} \) |
| 37 | \( 1 + 6.20T + 37T^{2} \) |
| 41 | \( 1 - 0.876T + 41T^{2} \) |
| 43 | \( 1 + 0.0269T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 2.16T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 5.33T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 - 5.03T + 71T^{2} \) |
| 73 | \( 1 - 4.45T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 8.71T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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
−8.110968873672707135676872051834, −7.05414468132883204990171264533, −6.69404602905690559937072337408, −5.78121518764757116633989137746, −5.25834651713598471866123264801, −4.48246688556013077106386444843, −3.70201295754067387089369760096, −2.97191318580120063776216690433, −2.37309705355400774326344515713, −0.57156561196420703576419892950,
0.57156561196420703576419892950, 2.37309705355400774326344515713, 2.97191318580120063776216690433, 3.70201295754067387089369760096, 4.48246688556013077106386444843, 5.25834651713598471866123264801, 5.78121518764757116633989137746, 6.69404602905690559937072337408, 7.05414468132883204990171264533, 8.110968873672707135676872051834