| L(s) = 1 | − 0.530·2-s + 0.899·3-s − 1.71·4-s + 3.03·5-s − 0.476·6-s + 0.743·7-s + 1.97·8-s − 2.19·9-s − 1.60·10-s + 3.91·11-s − 1.54·12-s − 1.48·13-s − 0.394·14-s + 2.72·15-s + 2.39·16-s + 5.45·17-s + 1.16·18-s + 1.78·19-s − 5.21·20-s + 0.668·21-s − 2.07·22-s − 4.14·23-s + 1.77·24-s + 4.19·25-s + 0.787·26-s − 4.66·27-s − 1.27·28-s + ⋯ |
| L(s) = 1 | − 0.374·2-s + 0.519·3-s − 0.859·4-s + 1.35·5-s − 0.194·6-s + 0.281·7-s + 0.697·8-s − 0.730·9-s − 0.508·10-s + 1.17·11-s − 0.446·12-s − 0.411·13-s − 0.105·14-s + 0.704·15-s + 0.597·16-s + 1.32·17-s + 0.273·18-s + 0.409·19-s − 1.16·20-s + 0.145·21-s − 0.442·22-s − 0.863·23-s + 0.361·24-s + 0.838·25-s + 0.154·26-s − 0.898·27-s − 0.241·28-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
| \(L(1)\) |
\(\approx\) |
\(1.208091814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208091814\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.530T + 2T^{2} \) |
| 3 | \( 1 - 0.899T + 3T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 - 0.743T + 7T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 + 4.14T + 23T^{2} \) |
| 29 | \( 1 + 4.67T + 29T^{2} \) |
| 31 | \( 1 + 0.723T + 31T^{2} \) |
| 37 | \( 1 + 0.460T + 37T^{2} \) |
| 41 | \( 1 + 6.33T + 41T^{2} \) |
| 43 | \( 1 - 0.907T + 43T^{2} \) |
| 47 | \( 1 + 2.75T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 - 5.55T + 61T^{2} \) |
| 67 | \( 1 + 1.92T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 6.59T + 79T^{2} \) |
| 83 | \( 1 + 2.20T + 83T^{2} \) |
| 89 | \( 1 + 5.90T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−12.65236875114677382241127525525, −11.48774094126597515522097754254, −9.995436354886708022464939945429, −9.563797734390071901165869257566, −8.705504178036617020568536648820, −7.70077236295669609096332332748, −6.08267796047963842289649758594, −5.13145926464621747028654331750, −3.50951263376479321404567945896, −1.69597385220128136931077491004,
1.69597385220128136931077491004, 3.50951263376479321404567945896, 5.13145926464621747028654331750, 6.08267796047963842289649758594, 7.70077236295669609096332332748, 8.705504178036617020568536648820, 9.563797734390071901165869257566, 9.995436354886708022464939945429, 11.48774094126597515522097754254, 12.65236875114677382241127525525
