| L(s) = 1 | − 19.8·2-s − 24.8·3-s + 265.·4-s − 146.·5-s + 493.·6-s − 946.·7-s − 2.72e3·8-s − 1.56e3·9-s + 2.91e3·10-s + 5.26e3·11-s − 6.59e3·12-s + 5.56e3·13-s + 1.87e4·14-s + 3.64e3·15-s + 2.00e4·16-s − 2.65e4·17-s + 3.11e4·18-s − 2.47e4·19-s − 3.89e4·20-s + 2.35e4·21-s − 1.04e5·22-s + 5.18e4·23-s + 6.76e4·24-s − 5.65e4·25-s − 1.10e5·26-s + 9.33e4·27-s − 2.50e5·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s − 0.531·3-s + 2.07·4-s − 0.525·5-s + 0.931·6-s − 1.04·7-s − 1.87·8-s − 0.717·9-s + 0.920·10-s + 1.19·11-s − 1.10·12-s + 0.702·13-s + 1.82·14-s + 0.279·15-s + 1.22·16-s − 1.31·17-s + 1.25·18-s − 0.826·19-s − 1.08·20-s + 0.554·21-s − 2.09·22-s + 0.887·23-s + 0.999·24-s − 0.724·25-s − 1.23·26-s + 0.913·27-s − 2.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 7.64e6T \) |
| good | 2 | \( 1 + 19.8T + 128T^{2} \) |
| 3 | \( 1 + 24.8T + 2.18e3T^{2} \) |
| 5 | \( 1 + 146.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 946.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.26e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.56e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.65e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.47e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.18e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.26e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.46e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.83e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.70e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.15e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.02e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.60e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.83e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.52e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.13e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.66e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.48e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.07e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.00e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.98e6T + 8.07e13T^{2} \) |
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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
−10.83950916607678979890719539360, −9.365345616332751008045423664182, −9.006307958744584381471020230703, −7.87775387178422506132510215751, −6.61890429364470564827784418246, −6.17400447343358657528251681388, −3.98481814972512739524226777321, −2.47489512883218816457522953866, −0.896760972295751264681932693150, 0,
0.896760972295751264681932693150, 2.47489512883218816457522953866, 3.98481814972512739524226777321, 6.17400447343358657528251681388, 6.61890429364470564827784418246, 7.87775387178422506132510215751, 9.006307958744584381471020230703, 9.365345616332751008045423664182, 10.83950916607678979890719539360
