| L(s) = 1 | − 42.9·2-s + 78.7·3-s + 1.33e3·4-s + 2.23e3·5-s − 3.38e3·6-s + 958.·7-s − 3.52e4·8-s − 1.34e4·9-s − 9.60e4·10-s + 3.04e4·11-s + 1.05e5·12-s + 1.60e4·13-s − 4.11e4·14-s + 1.76e5·15-s + 8.33e5·16-s + 5.96e5·17-s + 5.78e5·18-s − 8.34e5·19-s + 2.98e6·20-s + 7.55e4·21-s − 1.30e6·22-s − 1.56e6·23-s − 2.78e6·24-s + 3.04e6·25-s − 6.87e5·26-s − 2.61e6·27-s + 1.27e6·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.561·3-s + 2.60·4-s + 1.60·5-s − 1.06·6-s + 0.150·7-s − 3.04·8-s − 0.684·9-s − 3.03·10-s + 0.626·11-s + 1.46·12-s + 0.155·13-s − 0.286·14-s + 0.898·15-s + 3.17·16-s + 1.73·17-s + 1.29·18-s − 1.46·19-s + 4.16·20-s + 0.0847·21-s − 1.18·22-s − 1.16·23-s − 1.71·24-s + 1.56·25-s − 0.295·26-s − 0.946·27-s + 0.393·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + 1.50e9T \) |
| good | 2 | \( 1 + 42.9T + 512T^{2} \) |
| 3 | \( 1 - 78.7T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.23e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 958.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.04e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.60e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.96e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.56e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.26e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.73e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.30e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.96e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.48e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.59e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.56e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.86e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.83e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.90e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.42e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.15e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.56e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.85e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.17e9T + 7.60e17T^{2} \) |
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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
−9.875135698479995131413866252463, −9.454990563538288648450358976992, −8.532123877571841275489350473742, −7.75771880713616963171064615622, −6.36089133557433209159598559979, −5.77228250048644339424154430785, −3.21058126782501590355509344343, −2.00464481284006761161317912349, −1.52864183852920667690863383206, 0,
1.52864183852920667690863383206, 2.00464481284006761161317912349, 3.21058126782501590355509344343, 5.77228250048644339424154430785, 6.36089133557433209159598559979, 7.75771880713616963171064615622, 8.532123877571841275489350473742, 9.454990563538288648450358976992, 9.875135698479995131413866252463
