| L(s) = 1 | − 72.3·2-s − 558.·3-s + 3.17e3·4-s + 7.09e3·5-s + 4.03e4·6-s − 1.71e4·7-s − 8.18e4·8-s + 1.34e5·9-s − 5.13e5·10-s + 2.11e4·11-s − 1.77e6·12-s + 2.73e5·13-s + 1.24e6·14-s − 3.96e6·15-s − 5.94e5·16-s − 6.68e6·17-s − 9.74e6·18-s − 4.92e6·19-s + 2.25e7·20-s + 9.59e6·21-s − 1.52e6·22-s − 4.17e7·23-s + 4.57e7·24-s + 1.55e6·25-s − 1.97e7·26-s + 2.36e7·27-s − 5.46e7·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 1.32·3-s + 1.55·4-s + 1.01·5-s + 2.12·6-s − 0.386·7-s − 0.883·8-s + 0.761·9-s − 1.62·10-s + 0.0395·11-s − 2.06·12-s + 0.204·13-s + 0.617·14-s − 1.34·15-s − 0.141·16-s − 1.14·17-s − 1.21·18-s − 0.455·19-s + 1.57·20-s + 0.512·21-s − 0.0631·22-s − 1.35·23-s + 1.17·24-s + 0.0318·25-s − 0.326·26-s + 0.317·27-s − 0.600·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.2719553265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2719553265\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + 2.96e11T \) |
| good | 2 | \( 1 + 72.3T + 2.04e3T^{2} \) |
| 3 | \( 1 + 558.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 7.09e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 1.71e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.11e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.73e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.68e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 4.92e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.17e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 8.40e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.59e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.36e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 2.79e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.72e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.09e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.01e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 7.29e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 5.90e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.81e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 3.13e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.76e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.97e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.91e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 7.27e8T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.08e10T + 7.15e21T^{2} \) |
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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
−10.48431661130150254997692358506, −9.572296386487377769234843202827, −8.815177391523230103409216716645, −7.50994283324308469161014522681, −6.32709769653925462616904119622, −5.98300971281414143903157764164, −4.47782783415096160762944602597, −2.40165834945430271396442720787, −1.45837300604082864565120526554, −0.32759821222603320521478864405,
0.32759821222603320521478864405, 1.45837300604082864565120526554, 2.40165834945430271396442720787, 4.47782783415096160762944602597, 5.98300971281414143903157764164, 6.32709769653925462616904119622, 7.50994283324308469161014522681, 8.815177391523230103409216716645, 9.572296386487377769234843202827, 10.48431661130150254997692358506
