| L(s) = 1 | + (0.661 − 0.173i)2-s + (−0.901 − 0.767i)3-s + (−1.33 + 0.751i)4-s + (−2.05 + 2.25i)5-s + (−0.729 − 0.351i)6-s + (−2.04 − 0.535i)7-s + (−1.73 + 1.68i)8-s + (−0.255 − 1.57i)9-s + (−0.965 + 1.85i)10-s + (−0.971 + 2.19i)11-s + (1.78 + 0.346i)12-s + (−0.311 + 0.0401i)13-s − 1.44·14-s + (3.58 − 0.462i)15-s + (0.732 − 1.20i)16-s + (−1.70 − 0.109i)17-s + ⋯ |
| L(s) = 1 | + (0.467 − 0.122i)2-s + (−0.520 − 0.442i)3-s + (−0.667 + 0.375i)4-s + (−0.917 + 1.01i)5-s + (−0.297 − 0.143i)6-s + (−0.771 − 0.202i)7-s + (−0.613 + 0.594i)8-s + (−0.0850 − 0.526i)9-s + (−0.305 + 0.585i)10-s + (−0.292 + 0.661i)11-s + (0.513 + 0.100i)12-s + (−0.0863 + 0.0111i)13-s − 0.385·14-s + (0.925 − 0.119i)15-s + (0.183 − 0.302i)16-s + (−0.413 − 0.0265i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0852926 + 0.276580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0852926 + 0.276580i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (-14.0 - 0.430i)T \) |
| good | 2 | \( 1 + (-0.661 + 0.173i)T + (1.74 - 0.981i)T^{2} \) |
| 3 | \( 1 + (0.901 + 0.767i)T + (0.478 + 2.96i)T^{2} \) |
| 5 | \( 1 + (2.05 - 2.25i)T + (-0.480 - 4.97i)T^{2} \) |
| 7 | \( 1 + (2.04 + 0.535i)T + (6.09 + 3.43i)T^{2} \) |
| 11 | \( 1 + (0.971 - 2.19i)T + (-7.39 - 8.14i)T^{2} \) |
| 13 | \( 1 + (0.311 - 0.0401i)T + (12.5 - 3.29i)T^{2} \) |
| 17 | \( 1 + (1.70 + 0.109i)T + (16.8 + 2.17i)T^{2} \) |
| 19 | \( 1 + (-0.307 - 1.34i)T + (-17.1 + 8.24i)T^{2} \) |
| 23 | \( 1 + (-6.44 - 4.80i)T + (6.54 + 22.0i)T^{2} \) |
| 29 | \( 1 + (4.06 + 0.792i)T + (26.8 + 10.8i)T^{2} \) |
| 31 | \( 1 + (0.0946 + 2.95i)T + (-30.9 + 1.98i)T^{2} \) |
| 37 | \( 1 + (2.84 + 4.70i)T + (-17.1 + 32.8i)T^{2} \) |
| 41 | \( 1 + (11.4 + 0.733i)T + (40.6 + 5.24i)T^{2} \) |
| 43 | \( 1 + (3.85 - 8.69i)T + (-28.9 - 31.8i)T^{2} \) |
| 47 | \( 1 + (-1.83 + 4.97i)T + (-35.7 - 30.4i)T^{2} \) |
| 53 | \( 1 + (-8.97 - 3.63i)T + (38.0 + 36.8i)T^{2} \) |
| 59 | \( 1 + (-6.78 - 12.9i)T + (-33.7 + 48.3i)T^{2} \) |
| 61 | \( 1 + (5.87 - 4.99i)T + (9.73 - 60.2i)T^{2} \) |
| 67 | \( 1 + (0.845 - 2.29i)T + (-51.0 - 43.4i)T^{2} \) |
| 71 | \( 1 + (-0.959 - 5.93i)T + (-67.3 + 22.3i)T^{2} \) |
| 73 | \( 1 + (-3.22 - 5.31i)T + (-33.7 + 64.7i)T^{2} \) |
| 79 | \( 1 + (5.14 + 5.66i)T + (-7.58 + 78.6i)T^{2} \) |
| 83 | \( 1 + (-1.26 + 5.54i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (0.158 - 4.93i)T + (-88.8 - 5.70i)T^{2} \) |
| 97 | \( 1 + (1.87 + 2.69i)T + (-33.5 + 91.0i)T^{2} \) |
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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
−12.88154755961236706192103350215, −11.92237889525726199121395035208, −11.33274171971045186844288966830, −10.02869008665277034453016045995, −8.923589071844206392712191202490, −7.47010480586380823157746547702, −6.85148834577901455609457908254, −5.49430576660087265013413273551, −3.96429378268650754452004150908, −3.09958653089239399915351920742,
0.23042069631894926541971987883, 3.48349484763176800868797192571, 4.76918179801761432276918029564, 5.27730771255927516199225419268, 6.64552138901980804846252045160, 8.332219960689758822571303289905, 8.998137072545795229724821595430, 10.17922725202754483530225311298, 11.18169673959266012661028185898, 12.28278208203515960950940148298
