| L(s) = 1 | + 0.896·2-s + 3.41·3-s − 1.19·4-s − 0.448·5-s + 3.05·6-s − 2.39·7-s − 2.86·8-s + 8.64·9-s − 0.401·10-s + 3.36·11-s − 4.08·12-s − 3.92·13-s − 2.14·14-s − 1.53·15-s − 0.175·16-s + 0.536·17-s + 7.74·18-s − 3.95·19-s + 0.536·20-s − 8.16·21-s + 3.01·22-s − 5.71·23-s − 9.77·24-s − 4.79·25-s − 3.52·26-s + 19.2·27-s + 2.86·28-s + ⋯ |
| L(s) = 1 | + 0.633·2-s + 1.97·3-s − 0.598·4-s − 0.200·5-s + 1.24·6-s − 0.903·7-s − 1.01·8-s + 2.88·9-s − 0.127·10-s + 1.01·11-s − 1.17·12-s − 1.08·13-s − 0.572·14-s − 0.395·15-s − 0.0438·16-s + 0.130·17-s + 1.82·18-s − 0.908·19-s + 0.119·20-s − 1.78·21-s + 0.642·22-s − 1.19·23-s − 1.99·24-s − 0.959·25-s − 0.690·26-s + 3.70·27-s + 0.540·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.171149165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.171149165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - T \) |
| good | 2 | \( 1 - 0.896T + 2T^{2} \) |
| 3 | \( 1 - 3.41T + 3T^{2} \) |
| 5 | \( 1 + 0.448T + 5T^{2} \) |
| 7 | \( 1 + 2.39T + 7T^{2} \) |
| 11 | \( 1 - 3.36T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 - 0.536T + 17T^{2} \) |
| 19 | \( 1 + 3.95T + 19T^{2} \) |
| 23 | \( 1 + 5.71T + 23T^{2} \) |
| 29 | \( 1 + 6.05T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 - 1.84T + 47T^{2} \) |
| 53 | \( 1 - 9.84T + 53T^{2} \) |
| 59 | \( 1 + 4.25T + 59T^{2} \) |
| 61 | \( 1 + 0.391T + 61T^{2} \) |
| 67 | \( 1 - 7.54T + 67T^{2} \) |
| 71 | \( 1 + 8.07T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 2.38T + 79T^{2} \) |
| 83 | \( 1 + 1.38T + 83T^{2} \) |
| 89 | \( 1 + 1.54T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−12.82957803084216534502116201369, −12.06378931200110123501258261716, −9.848923205750769912872849113672, −9.569164464581891717710623859414, −8.568977710575794230809824739828, −7.60873228653269623285234981291, −6.32527951321528156948281656717, −4.32346730235717361093642779739, −3.71933729323563848878574036472, −2.47236704046632529570457532303,
2.47236704046632529570457532303, 3.71933729323563848878574036472, 4.32346730235717361093642779739, 6.32527951321528156948281656717, 7.60873228653269623285234981291, 8.568977710575794230809824739828, 9.569164464581891717710623859414, 9.848923205750769912872849113672, 12.06378931200110123501258261716, 12.82957803084216534502116201369
