| L(s) = 1 | + (−0.375 − 2.32i)2-s + (0.220 + 0.242i)3-s + (−3.36 + 1.11i)4-s + (−4.10 + 0.529i)5-s + (0.480 − 0.602i)6-s + (−0.687 + 4.25i)7-s + (1.68 + 3.23i)8-s + (0.277 − 2.87i)9-s + (2.77 + 9.35i)10-s + (−2.04 − 0.131i)11-s + (−1.01 − 0.570i)12-s + (0.928 + 0.790i)13-s + 10.1·14-s + (−1.03 − 0.878i)15-s + (1.20 − 0.896i)16-s + (−1.03 − 2.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.265 − 1.64i)2-s + (0.127 + 0.139i)3-s + (−1.68 + 0.559i)4-s + (−1.83 + 0.236i)5-s + (0.196 − 0.246i)6-s + (−0.259 + 1.60i)7-s + (0.596 + 1.14i)8-s + (0.0925 − 0.959i)9-s + (0.877 + 2.95i)10-s + (−0.615 − 0.0395i)11-s + (−0.292 − 0.164i)12-s + (0.257 + 0.219i)13-s + 2.71·14-s + (−0.266 − 0.226i)15-s + (0.300 − 0.224i)16-s + (−0.249 − 0.679i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0305 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0305 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0146190 + 0.0141793i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0146190 + 0.0141793i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (13.9 + 1.87i)T \) |
| good | 2 | \( 1 + (0.375 + 2.32i)T + (-1.89 + 0.630i)T^{2} \) |
| 3 | \( 1 + (-0.220 - 0.242i)T + (-0.288 + 2.98i)T^{2} \) |
| 5 | \( 1 + (4.10 - 0.529i)T + (4.83 - 1.26i)T^{2} \) |
| 7 | \( 1 + (0.687 - 4.25i)T + (-6.64 - 2.20i)T^{2} \) |
| 11 | \( 1 + (2.04 + 0.131i)T + (10.9 + 1.40i)T^{2} \) |
| 13 | \( 1 + (-0.928 - 0.790i)T + (2.07 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.03 + 2.80i)T + (-12.9 + 11.0i)T^{2} \) |
| 19 | \( 1 + (6.57 + 3.16i)T + (11.8 + 14.8i)T^{2} \) |
| 23 | \( 1 + (4.12 - 1.66i)T + (16.5 - 16.0i)T^{2} \) |
| 29 | \( 1 + (0.450 + 0.253i)T + (15.0 + 24.7i)T^{2} \) |
| 31 | \( 1 + (-2.44 - 3.50i)T + (-10.7 + 29.0i)T^{2} \) |
| 37 | \( 1 + (-1.53 - 1.14i)T + (10.5 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-1.18 - 3.21i)T + (-31.2 + 26.5i)T^{2} \) |
| 43 | \( 1 + (6.11 + 0.392i)T + (42.6 + 5.49i)T^{2} \) |
| 47 | \( 1 + (0.0126 - 0.0284i)T + (-31.5 - 34.7i)T^{2} \) |
| 53 | \( 1 + (7.03 + 11.5i)T + (-24.5 + 46.9i)T^{2} \) |
| 59 | \( 1 + (1.46 - 4.94i)T + (-49.4 - 32.1i)T^{2} \) |
| 61 | \( 1 + (-2.74 + 3.02i)T + (-5.85 - 60.7i)T^{2} \) |
| 67 | \( 1 + (0.00351 - 0.00793i)T + (-45.0 - 49.5i)T^{2} \) |
| 71 | \( 1 + (0.893 - 9.25i)T + (-69.6 - 13.5i)T^{2} \) |
| 73 | \( 1 + (-9.09 - 6.78i)T + (20.7 + 69.9i)T^{2} \) |
| 79 | \( 1 + (17.1 + 2.21i)T + (76.4 + 20.0i)T^{2} \) |
| 83 | \( 1 + (-6.92 + 3.33i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-1.78 + 2.55i)T + (-30.7 - 83.5i)T^{2} \) |
| 97 | \( 1 + (5.45 - 3.55i)T + (39.2 - 88.6i)T^{2} \) |
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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
−12.33466201450057084952961427872, −11.67088390061573607818943649855, −11.17572081898359195418656695967, −9.897823267320403774522912330747, −8.817883462324472848064119976699, −8.320709441374417311315262973876, −6.60600646553523494730281445215, −4.62747082538380709781393052763, −3.52307744550557911905265389325, −2.59526075004943595176512177735,
0.01864561737587065103422834263, 3.97088074403857039398515156673, 4.64250446989531780945416706767, 6.35776559310999878439271471091, 7.46147651321945821740280285904, 7.911150327228490146284148590812, 8.487430465425033291527351075917, 10.30572743515295201315581078950, 11.01381883333191991884855649530, 12.59992474258052895559138752096
