| L(s) = 1 | + 0.308·2-s + 3-s − 1.90·4-s + 2.48·5-s + 0.308·6-s + 3.17·7-s − 1.20·8-s + 9-s + 0.765·10-s − 2.66·11-s − 1.90·12-s + 6.37·13-s + 0.978·14-s + 2.48·15-s + 3.43·16-s + 17-s + 0.308·18-s + 8.52·19-s − 4.72·20-s + 3.17·21-s − 0.822·22-s + 6.90·23-s − 1.20·24-s + 1.16·25-s + 1.96·26-s + 27-s − 6.04·28-s + ⋯ |
| L(s) = 1 | + 0.217·2-s + 0.577·3-s − 0.952·4-s + 1.11·5-s + 0.125·6-s + 1.20·7-s − 0.425·8-s + 0.333·9-s + 0.242·10-s − 0.804·11-s − 0.549·12-s + 1.76·13-s + 0.261·14-s + 0.641·15-s + 0.859·16-s + 0.242·17-s + 0.0726·18-s + 1.95·19-s − 1.05·20-s + 0.692·21-s − 0.175·22-s + 1.43·23-s − 0.245·24-s + 0.232·25-s + 0.385·26-s + 0.192·27-s − 1.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.406393155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.406393155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 0.308T + 2T^{2} \) |
| 5 | \( 1 - 2.48T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 - 6.37T + 13T^{2} \) |
| 19 | \( 1 - 8.52T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 29 | \( 1 + 2.52T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 1.66T + 41T^{2} \) |
| 43 | \( 1 + 1.09T + 43T^{2} \) |
| 47 | \( 1 - 3.69T + 47T^{2} \) |
| 53 | \( 1 + 0.219T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 4.77T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 6.68T + 97T^{2} \) |
| 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
−8.716681742142359895240910840307, −7.78813090126448336529812530859, −7.19016113118203639915753456358, −5.84873802132001555601248972769, −5.37568046673171427264159710817, −4.91479620341529252830049884217, −3.69112939839868111288315577398, −3.16201413700836099058924049943, −1.81078263166900856976502893480, −1.14185815515325249597961012458,
1.14185815515325249597961012458, 1.81078263166900856976502893480, 3.16201413700836099058924049943, 3.69112939839868111288315577398, 4.91479620341529252830049884217, 5.37568046673171427264159710817, 5.84873802132001555601248972769, 7.19016113118203639915753456358, 7.78813090126448336529812530859, 8.716681742142359895240910840307
