| L(s) = 1 | + 9.73·2-s − 22.3·3-s + 62.8·4-s − 58.5·5-s − 217.·6-s − 2.91·7-s + 300.·8-s + 254.·9-s − 570.·10-s − 29.1·11-s − 1.40e3·12-s + 771.·13-s − 28.3·14-s + 1.30e3·15-s + 916.·16-s + 1.05e3·17-s + 2.48e3·18-s + 668.·19-s − 3.68e3·20-s + 65.0·21-s − 283.·22-s + 1.20e3·23-s − 6.70e3·24-s + 302.·25-s + 7.51e3·26-s − 267.·27-s − 183.·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 1.43·3-s + 1.96·4-s − 1.04·5-s − 2.46·6-s − 0.0224·7-s + 1.66·8-s + 1.04·9-s − 1.80·10-s − 0.0725·11-s − 2.81·12-s + 1.26·13-s − 0.0386·14-s + 1.49·15-s + 0.894·16-s + 0.885·17-s + 1.80·18-s + 0.424·19-s − 2.05·20-s + 0.0321·21-s − 0.124·22-s + 0.474·23-s − 2.37·24-s + 0.0967·25-s + 2.18·26-s − 0.0706·27-s − 0.0441·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.984354503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.984354503\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 3.88e4T \) |
| good | 2 | \( 1 - 9.73T + 32T^{2} \) |
| 3 | \( 1 + 22.3T + 243T^{2} \) |
| 5 | \( 1 + 58.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 2.91T + 1.68e4T^{2} \) |
| 11 | \( 1 + 29.1T + 1.61e5T^{2} \) |
| 13 | \( 1 - 771.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 668.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.20e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.06e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 688.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 85.4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 190.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.73e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.46e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.91e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.42e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.06e4T + 8.58e9T^{2} \) |
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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
−11.72207136454237611381735895657, −11.30185599071021561080008673998, −10.28473824431592159581006752647, −8.195425047247561596383396246553, −6.88600583517086732224128143278, −6.08288044721032318197316201221, −5.16196647161861787893410082738, −4.22942047272726962207822934453, −3.18122054267582537735511141592, −0.920127943011961056368696424143,
0.920127943011961056368696424143, 3.18122054267582537735511141592, 4.22942047272726962207822934453, 5.16196647161861787893410082738, 6.08288044721032318197316201221, 6.88600583517086732224128143278, 8.195425047247561596383396246553, 10.28473824431592159581006752647, 11.30185599071021561080008673998, 11.72207136454237611381735895657
