| L(s) = 1 | − 5.80·2-s + 12.9·3-s + 1.67·4-s + 23.4·5-s − 74.9·6-s − 167.·7-s + 175.·8-s − 76.3·9-s − 136.·10-s − 355.·11-s + 21.6·12-s + 481.·13-s + 970.·14-s + 303.·15-s − 1.07e3·16-s − 535.·17-s + 443.·18-s + 1.41e3·19-s + 39.3·20-s − 2.15e3·21-s + 2.06e3·22-s + 4.28e3·23-s + 2.27e3·24-s − 2.57e3·25-s − 2.79e3·26-s − 4.12e3·27-s − 280.·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.828·3-s + 0.0523·4-s + 0.420·5-s − 0.849·6-s − 1.29·7-s + 0.972·8-s − 0.314·9-s − 0.431·10-s − 0.885·11-s + 0.0433·12-s + 0.789·13-s + 1.32·14-s + 0.348·15-s − 1.04·16-s − 0.449·17-s + 0.322·18-s + 0.897·19-s + 0.0220·20-s − 1.06·21-s + 0.908·22-s + 1.68·23-s + 0.805·24-s − 0.823·25-s − 0.810·26-s − 1.08·27-s − 0.0675·28-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
| \(L(3)\) |
\(\approx\) |
\(1.075372808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.075372808\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 5.80T + 32T^{2} \) |
| 3 | \( 1 - 12.9T + 243T^{2} \) |
| 5 | \( 1 - 23.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 167.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 355.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 481.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 535.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.41e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.74e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.64e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.67e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e4T + 8.58e9T^{2} \) |
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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
−11.24074646506951728124456638109, −10.23147882985160439998132336584, −9.420846907912336575588336290417, −8.819490346614092040378190775726, −7.87139663901265104017585479280, −6.73688005487510798961327257291, −5.34480076198986330860008276516, −3.55656077281658554682419897809, −2.43776094813388608696629441808, −0.70819644205491908138051422173,
0.70819644205491908138051422173, 2.43776094813388608696629441808, 3.55656077281658554682419897809, 5.34480076198986330860008276516, 6.73688005487510798961327257291, 7.87139663901265104017585479280, 8.819490346614092040378190775726, 9.420846907912336575588336290417, 10.23147882985160439998132336584, 11.24074646506951728124456638109
