| L(s) = 1 | − 2.61·2-s + 3-s + 4.85·4-s − 2.61·6-s − 7.47·8-s + 9-s − 5.47·11-s + 4.85·12-s + 0.763·13-s + 9.85·16-s − 7.70·17-s − 2.61·18-s + 3.23·19-s + 14.3·22-s − 5·23-s − 7.47·24-s − 2·26-s + 27-s + 4.70·29-s − 4.47·31-s − 10.8·32-s − 5.47·33-s + 20.1·34-s + 4.85·36-s − 5.47·37-s − 8.47·38-s + 0.763·39-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.42·4-s − 1.06·6-s − 2.64·8-s + 0.333·9-s − 1.64·11-s + 1.40·12-s + 0.211·13-s + 2.46·16-s − 1.86·17-s − 0.617·18-s + 0.742·19-s + 3.05·22-s − 1.04·23-s − 1.52·24-s − 0.392·26-s + 0.192·27-s + 0.874·29-s − 0.803·31-s − 1.91·32-s − 0.952·33-s + 3.46·34-s + 0.809·36-s − 0.899·37-s − 1.37·38-s + 0.122·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6274580068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6274580068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 8.23T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 0.763T + 59T^{2} \) |
| 61 | \( 1 - 15.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 0.527T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 1.76T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 5.70T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.563362030179154189249851116051, −8.005408716677935806331482791598, −7.37280407324740512629002937130, −6.74483141747641033482288038539, −5.84504124464702679918995066724, −4.79122456603031799517233706677, −3.48663380880317772519994741935, −2.42488400560722441349749799093, −2.03701692734220149177413444035, −0.56063986540561002413308348445,
0.56063986540561002413308348445, 2.03701692734220149177413444035, 2.42488400560722441349749799093, 3.48663380880317772519994741935, 4.79122456603031799517233706677, 5.84504124464702679918995066724, 6.74483141747641033482288038539, 7.37280407324740512629002937130, 8.005408716677935806331482791598, 8.563362030179154189249851116051
