L(s) = 1 | + 2.18·2-s − 1.75·3-s + 2.76·4-s − 2.11·5-s − 3.83·6-s + 1.67·8-s + 0.0920·9-s − 4.60·10-s − 5.76·11-s − 4.86·12-s − 13-s + 3.71·15-s − 1.87·16-s − 1.64·17-s + 0.200·18-s + 2.67·19-s − 5.83·20-s − 12.5·22-s + 6.42·23-s − 2.94·24-s − 0.545·25-s − 2.18·26-s + 5.11·27-s − 6.04·29-s + 8.10·30-s − 5.12·31-s − 7.45·32-s + ⋯ |
L(s) = 1 | + 1.54·2-s − 1.01·3-s + 1.38·4-s − 0.943·5-s − 1.56·6-s + 0.591·8-s + 0.0306·9-s − 1.45·10-s − 1.73·11-s − 1.40·12-s − 0.277·13-s + 0.958·15-s − 0.469·16-s − 0.397·17-s + 0.0473·18-s + 0.612·19-s − 1.30·20-s − 2.68·22-s + 1.33·23-s − 0.600·24-s − 0.109·25-s − 0.428·26-s + 0.984·27-s − 1.12·29-s + 1.47·30-s − 0.919·31-s − 1.31·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 17 | \( 1 + 1.64T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 23 | \( 1 - 6.42T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 5.74T + 37T^{2} \) |
| 41 | \( 1 - 7.14T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 - 7.74T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 + 4.96T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
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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
−10.92253796983778188378654767793, −9.416295901542218913843704962640, −8.000060852260740647473787241149, −7.23485821734285162001561526069, −6.18712657887960414555232244686, −5.21169769279296351147468640058, −4.90408695605158104221164682534, −3.63280280255996636564836170294, −2.63505725218615570184100183145, 0,
2.63505725218615570184100183145, 3.63280280255996636564836170294, 4.90408695605158104221164682534, 5.21169769279296351147468640058, 6.18712657887960414555232244686, 7.23485821734285162001561526069, 8.000060852260740647473787241149, 9.416295901542218913843704962640, 10.92253796983778188378654767793