| L(s) = 1 | + (−2.90 − 0.0465i)2-s + (−0.438 + 0.107i)3-s + (4.41 + 0.141i)4-s + (−1.33 − 9.17i)5-s + (1.27 − 0.291i)6-s + (6.15 − 6.35i)7-s + (−1.21 − 0.0582i)8-s + (−7.79 + 4.06i)9-s + (3.44 + 26.6i)10-s + (15.1 + 7.61i)11-s + (−1.95 + 0.412i)12-s + (−17.2 − 7.29i)13-s + (−18.1 + 18.1i)14-s + (1.57 + 3.88i)15-s + (−14.1 − 0.906i)16-s + (12.9 − 8.73i)17-s + ⋯ |
| L(s) = 1 | + (−1.45 − 0.0232i)2-s + (−0.146 + 0.0358i)3-s + (1.10 + 0.0354i)4-s + (−0.266 − 1.83i)5-s + (0.213 − 0.0486i)6-s + (0.878 − 0.907i)7-s + (−0.151 − 0.00728i)8-s + (−0.866 + 0.452i)9-s + (0.344 + 2.66i)10-s + (1.38 + 0.692i)11-s + (−0.162 + 0.0344i)12-s + (−1.32 − 0.561i)13-s + (−1.29 + 1.29i)14-s + (0.104 + 0.258i)15-s + (−0.882 − 0.0566i)16-s + (0.762 − 0.513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.100545 - 0.467656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100545 - 0.467656i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (-77.8 + 180. i)T \) |
| good | 2 | \( 1 + (2.90 + 0.0465i)T + (3.99 + 0.128i)T^{2} \) |
| 3 | \( 1 + (0.438 - 0.107i)T + (7.97 - 4.16i)T^{2} \) |
| 5 | \( 1 + (1.33 + 9.17i)T + (-23.9 + 7.11i)T^{2} \) |
| 7 | \( 1 + (-6.15 + 6.35i)T + (-1.57 - 48.9i)T^{2} \) |
| 11 | \( 1 + (-15.1 - 7.61i)T + (72.3 + 96.9i)T^{2} \) |
| 13 | \( 1 + (17.2 + 7.29i)T + (117. + 121. i)T^{2} \) |
| 17 | \( 1 + (-12.9 + 8.73i)T + (108. - 267. i)T^{2} \) |
| 19 | \( 1 + (-4.74 + 3.78i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (-6.08 + 16.5i)T + (-402. - 342. i)T^{2} \) |
| 29 | \( 1 + (33.2 + 21.6i)T + (340. + 769. i)T^{2} \) |
| 31 | \( 1 + (19.0 - 23.1i)T + (-183. - 943. i)T^{2} \) |
| 37 | \( 1 + (50.1 - 3.21i)T + (1.35e3 - 175. i)T^{2} \) |
| 41 | \( 1 + (-8.45 + 43.4i)T + (-1.55e3 - 630. i)T^{2} \) |
| 43 | \( 1 + (-12.7 - 38.4i)T + (-1.48e3 + 1.10e3i)T^{2} \) |
| 47 | \( 1 + (-12.8 - 22.8i)T + (-1.14e3 + 1.88e3i)T^{2} \) |
| 53 | \( 1 + (-9.32 - 21.0i)T + (-1.88e3 + 2.07e3i)T^{2} \) |
| 59 | \( 1 + (60.4 + 7.79i)T + (3.36e3 + 882. i)T^{2} \) |
| 61 | \( 1 + (4.06 + 6.70i)T + (-1.72e3 + 3.29e3i)T^{2} \) |
| 67 | \( 1 + (-38.0 - 10.6i)T + (3.83e3 + 2.32e3i)T^{2} \) |
| 71 | \( 1 + (34.9 + 10.9i)T + (4.13e3 + 2.88e3i)T^{2} \) |
| 73 | \( 1 + (33.6 + 29.5i)T + (681. + 5.28e3i)T^{2} \) |
| 79 | \( 1 + (-9.25 + 63.7i)T + (-5.98e3 - 1.77e3i)T^{2} \) |
| 83 | \( 1 + (-24.6 - 19.6i)T + (1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-55.7 - 67.6i)T + (-1.51e3 + 7.77e3i)T^{2} \) |
| 97 | \( 1 + (-9.92 - 37.8i)T + (-8.19e3 + 4.61e3i)T^{2} \) |
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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
−11.75185922683394518536736989327, −10.67796000847605185933104874093, −9.525763852231340130551574801297, −8.938178948699795023411157760943, −7.913733930292503413162390412781, −7.34552642661217432038368299151, −5.21726181063776218413809998886, −4.42312937803580897615095806133, −1.62996000947607112229369180198, −0.46305368374877214203514772884,
1.98518867814011711574755662911, 3.46375942404978758136864825325, 5.73289790247313947082321444029, 6.87467405928019205133674872836, 7.66850274690673118157576118306, 8.810716104780362523741008199417, 9.583877002847192271971100655812, 10.73235882748992247167608193418, 11.56957712675361999912912459169, 11.83515950567142780942179151131
