| L(s) = 1 | − 10.7·2-s + 19.4·3-s + 82.7·4-s + 86.8·5-s − 208.·6-s − 107.·7-s − 543.·8-s + 136.·9-s − 930.·10-s + 73.1·11-s + 1.61e3·12-s − 497.·13-s + 1.15e3·14-s + 1.69e3·15-s + 3.17e3·16-s + 368.·17-s − 1.46e3·18-s + 976.·19-s + 7.18e3·20-s − 2.10e3·21-s − 783.·22-s + 4.09e3·23-s − 1.05e4·24-s + 4.41e3·25-s + 5.33e3·26-s − 2.06e3·27-s − 8.92e3·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 1.25·3-s + 2.58·4-s + 1.55·5-s − 2.36·6-s − 0.832·7-s − 3.00·8-s + 0.563·9-s − 2.94·10-s + 0.182·11-s + 3.23·12-s − 0.816·13-s + 1.57·14-s + 1.94·15-s + 3.10·16-s + 0.309·17-s − 1.06·18-s + 0.620·19-s + 4.01·20-s − 1.04·21-s − 0.345·22-s + 1.61·23-s − 3.75·24-s + 1.41·25-s + 1.54·26-s − 0.546·27-s − 2.15·28-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
| \(L(3)\) |
\(\approx\) |
\(1.628459032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628459032\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 10.7T + 32T^{2} \) |
| 3 | \( 1 - 19.4T + 243T^{2} \) |
| 5 | \( 1 - 86.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 107.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 73.1T + 1.61e5T^{2} \) |
| 13 | \( 1 + 497.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 368.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 976.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.44e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.39e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.18e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.71e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.58e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.97e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.36e5T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.07506173113465435809700907094, −9.933051274225912322068738873480, −9.456392466739912960429343915045, −9.052068512871726627414031971541, −7.81245969078994898086970680693, −6.89635790226067647775830606773, −5.80349582151469204699240657679, −2.93753304344969174342694365815, −2.33241885369118099625058241868, −1.01066128037433157390832138485,
1.01066128037433157390832138485, 2.33241885369118099625058241868, 2.93753304344969174342694365815, 5.80349582151469204699240657679, 6.89635790226067647775830606773, 7.81245969078994898086970680693, 9.052068512871726627414031971541, 9.456392466739912960429343915045, 9.933051274225912322068738873480, 11.07506173113465435809700907094
