| L(s) = 1 | + 1.81·2-s − 1.51·3-s + 1.28·4-s − 5-s − 2.75·6-s + 0.920·7-s − 1.28·8-s − 0.699·9-s − 1.81·10-s − 11-s − 1.95·12-s + 0.250·13-s + 1.66·14-s + 1.51·15-s − 4.91·16-s + 6.67·17-s − 1.26·18-s + 3.58·19-s − 1.28·20-s − 1.39·21-s − 1.81·22-s + 6.91·23-s + 1.95·24-s + 25-s + 0.454·26-s + 5.61·27-s + 1.18·28-s + ⋯ |
| L(s) = 1 | + 1.28·2-s − 0.875·3-s + 0.644·4-s − 0.447·5-s − 1.12·6-s + 0.347·7-s − 0.455·8-s − 0.233·9-s − 0.573·10-s − 0.301·11-s − 0.564·12-s + 0.0695·13-s + 0.446·14-s + 0.391·15-s − 1.22·16-s + 1.61·17-s − 0.298·18-s + 0.823·19-s − 0.288·20-s − 0.304·21-s − 0.386·22-s + 1.44·23-s + 0.399·24-s + 0.200·25-s + 0.0891·26-s + 1.07·27-s + 0.224·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 3 | \( 1 + 1.51T + 3T^{2} \) |
| 7 | \( 1 - 0.920T + 7T^{2} \) |
| 13 | \( 1 - 0.250T + 13T^{2} \) |
| 17 | \( 1 - 6.67T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 + 4.01T + 29T^{2} \) |
| 31 | \( 1 + 7.27T + 31T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 - 1.44T + 47T^{2} \) |
| 53 | \( 1 + 7.54T + 53T^{2} \) |
| 59 | \( 1 + 1.68T + 59T^{2} \) |
| 61 | \( 1 + 4.36T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 79 | \( 1 + 0.231T + 79T^{2} \) |
| 83 | \( 1 - 1.54T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.86669046923157869326461279707, −7.15795066395786348755052339060, −6.32243789602724784471905286560, −5.47106189542984597691811564897, −5.23593235166085690866343993389, −4.51980722939811949946745881156, −3.33310590388510130436812569891, −3.08417948752984344944307398235, −1.41656115065115877054740635169, 0,
1.41656115065115877054740635169, 3.08417948752984344944307398235, 3.33310590388510130436812569891, 4.51980722939811949946745881156, 5.23593235166085690866343993389, 5.47106189542984597691811564897, 6.32243789602724784471905286560, 7.15795066395786348755052339060, 7.86669046923157869326461279707
