| L(s) = 1 | − 2.24·2-s − 1.99·3-s + 3.05·4-s − 0.623·5-s + 4.48·6-s + 4.70·7-s − 2.37·8-s + 0.973·9-s + 1.40·10-s + 11-s − 6.09·12-s + 0.567·13-s − 10.5·14-s + 1.24·15-s − 0.768·16-s − 3.70·17-s − 2.19·18-s − 5.06·19-s − 1.90·20-s − 9.37·21-s − 2.24·22-s − 4.25·23-s + 4.73·24-s − 4.61·25-s − 1.27·26-s + 4.03·27-s + 14.3·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 1.15·3-s + 1.52·4-s − 0.278·5-s + 1.83·6-s + 1.77·7-s − 0.840·8-s + 0.324·9-s + 0.443·10-s + 0.301·11-s − 1.75·12-s + 0.157·13-s − 2.82·14-s + 0.320·15-s − 0.192·16-s − 0.899·17-s − 0.516·18-s − 1.16·19-s − 0.426·20-s − 2.04·21-s − 0.479·22-s − 0.886·23-s + 0.967·24-s − 0.922·25-s − 0.250·26-s + 0.777·27-s + 2.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 + 1.99T + 3T^{2} \) |
| 5 | \( 1 + 0.623T + 5T^{2} \) |
| 7 | \( 1 - 4.70T + 7T^{2} \) |
| 13 | \( 1 - 0.567T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 23 | \( 1 + 4.25T + 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 + 5.67T + 53T^{2} \) |
| 59 | \( 1 + 8.80T + 59T^{2} \) |
| 61 | \( 1 + 2.26T + 61T^{2} \) |
| 67 | \( 1 - 9.21T + 67T^{2} \) |
| 71 | \( 1 + 7.14T + 71T^{2} \) |
| 73 | \( 1 - 7.84T + 73T^{2} \) |
| 79 | \( 1 - 0.313T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.923729495455077204961255665869, −8.314328641534309231555505995170, −7.87140456546537266448453671498, −6.75661029616068327182954402249, −6.15713746617803246984120883381, −4.91383559235669391557228625714, −4.30998944910199053920576319704, −2.23660881061781519599786149683, −1.30857123071340237962804891555, 0,
1.30857123071340237962804891555, 2.23660881061781519599786149683, 4.30998944910199053920576319704, 4.91383559235669391557228625714, 6.15713746617803246984120883381, 6.75661029616068327182954402249, 7.87140456546537266448453671498, 8.314328641534309231555505995170, 8.923729495455077204961255665869
