| L(s) = 1 | + (0.532 + 0.0304i)2-s + (−0.448 + 1.37i)3-s + (−7.66 − 0.879i)4-s + (−3.29 − 6.24i)5-s + (−0.280 + 0.720i)6-s + (17.8 + 7.55i)7-s + (−8.25 − 1.42i)8-s + (20.1 + 14.6i)9-s + (−1.56 − 3.42i)10-s + (36.3 + 3.58i)11-s + (4.64 − 10.1i)12-s + (45.5 − 25.7i)13-s + (9.28 + 4.56i)14-s + (10.0 − 1.74i)15-s + (55.7 + 12.9i)16-s + (−0.967 + 0.345i)17-s + ⋯ |
| L(s) = 1 | + (0.188 + 0.0107i)2-s + (−0.0862 + 0.265i)3-s + (−0.958 − 0.109i)4-s + (−0.294 − 0.558i)5-s + (−0.0190 + 0.0490i)6-s + (0.965 + 0.407i)7-s + (−0.364 − 0.0631i)8-s + (0.745 + 0.541i)9-s + (−0.0494 − 0.108i)10-s + (0.995 + 0.0982i)11-s + (0.111 − 0.244i)12-s + (0.970 − 0.548i)13-s + (0.177 + 0.0871i)14-s + (0.173 − 0.0300i)15-s + (0.871 + 0.202i)16-s + (−0.0138 + 0.00492i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.59216 + 0.191401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59216 + 0.191401i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-36.3 - 3.58i)T \) |
| good | 2 | \( 1 + (-0.532 - 0.0304i)T + (7.94 + 0.911i)T^{2} \) |
| 3 | \( 1 + (0.448 - 1.37i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (3.29 + 6.24i)T + (-70.5 + 103. i)T^{2} \) |
| 7 | \( 1 + (-17.8 - 7.55i)T + (239. + 245. i)T^{2} \) |
| 13 | \( 1 + (-45.5 + 25.7i)T + (1.13e3 - 1.88e3i)T^{2} \) |
| 17 | \( 1 + (0.967 - 0.345i)T + (3.80e3 - 3.10e3i)T^{2} \) |
| 19 | \( 1 + (2.82 - 98.8i)T + (-6.84e3 - 391. i)T^{2} \) |
| 23 | \( 1 + (60.7 + 70.1i)T + (-1.73e3 + 1.20e4i)T^{2} \) |
| 29 | \( 1 + (-51.0 - 74.6i)T + (-8.84e3 + 2.27e4i)T^{2} \) |
| 31 | \( 1 + (0.989 - 4.88i)T + (-2.74e4 - 1.15e4i)T^{2} \) |
| 37 | \( 1 + (-243. + 223. i)T + (4.33e3 - 5.04e4i)T^{2} \) |
| 41 | \( 1 + (-56.2 - 213. i)T + (-6.00e4 + 3.38e4i)T^{2} \) |
| 43 | \( 1 + (-11.3 - 78.9i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + (-37.4 + 30.5i)T + (2.06e4 - 1.01e5i)T^{2} \) |
| 53 | \( 1 + (394. - 91.6i)T + (1.33e5 - 6.56e4i)T^{2} \) |
| 59 | \( 1 + (10.5 - 40.2i)T + (-1.78e5 - 1.00e5i)T^{2} \) |
| 61 | \( 1 + (-280. + 16.0i)T + (2.25e5 - 2.58e4i)T^{2} \) |
| 67 | \( 1 + (494. - 317. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-447. + 580. i)T + (-9.09e4 - 3.46e5i)T^{2} \) |
| 73 | \( 1 + (-139. + 231. i)T + (-1.81e5 - 3.44e5i)T^{2} \) |
| 79 | \( 1 + (787. - 810. i)T + (-1.40e4 - 4.92e5i)T^{2} \) |
| 83 | \( 1 + (10.4 - 121. i)T + (-5.63e5 - 9.75e4i)T^{2} \) |
| 89 | \( 1 + (77.9 + 22.8i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-73.3 + 139. i)T + (-5.15e5 - 7.53e5i)T^{2} \) |
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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
−12.93844463395268677268767433283, −12.19933534113210629146445626696, −10.93627474432743823166085215335, −9.805514755714348198271489713352, −8.638890675190943516854086680300, −7.974999809936235592807734286738, −5.98483437528666586364393033392, −4.76070251531698061795657812700, −3.97407195293457574739287495142, −1.29517875785097956995564487920,
1.15678550191555831610421666161, 3.69372551545665909197737137520, 4.58301550189430792981637332014, 6.31489441072347323806642571822, 7.45582739847166438191215628604, 8.686066255342743844423459977085, 9.662822164485466661017369240231, 11.11056809380951288814677655394, 11.83892735093463514853553373355, 13.12750538809020693947533420361
