| L(s) = 1 | + (0.425 − 2.96i)2-s + (−1.87 − 4.10i)3-s + (−0.915 − 0.268i)4-s + (12.1 + 14.0i)5-s + (−12.9 + 3.80i)6-s + (−5.88 − 3.78i)7-s + (8.75 − 19.1i)8-s + (4.33 − 5.00i)9-s + (46.8 − 30.1i)10-s + (−2.19 − 15.2i)11-s + (0.612 + 4.26i)12-s + (8.96 − 5.76i)13-s + (−13.7 + 15.8i)14-s + (34.9 − 76.4i)15-s + (−59.4 − 38.2i)16-s + (125. − 36.9i)17-s + ⋯ |
| L(s) = 1 | + (0.150 − 1.04i)2-s + (−0.360 − 0.790i)3-s + (−0.114 − 0.0336i)4-s + (1.09 + 1.25i)5-s + (−0.881 + 0.258i)6-s + (−0.317 − 0.204i)7-s + (0.387 − 0.847i)8-s + (0.160 − 0.185i)9-s + (1.48 − 0.952i)10-s + (−0.0601 − 0.418i)11-s + (0.0147 + 0.102i)12-s + (0.191 − 0.122i)13-s + (−0.261 + 0.302i)14-s + (0.600 − 1.31i)15-s + (−0.929 − 0.597i)16-s + (1.79 − 0.527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 + 0.890i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.12273 - 1.83341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12273 - 1.83341i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (5.88 + 3.78i)T \) |
| 23 | \( 1 + (87.7 + 66.8i)T \) |
| good | 2 | \( 1 + (-0.425 + 2.96i)T + (-7.67 - 2.25i)T^{2} \) |
| 3 | \( 1 + (1.87 + 4.10i)T + (-17.6 + 20.4i)T^{2} \) |
| 5 | \( 1 + (-12.1 - 14.0i)T + (-17.7 + 123. i)T^{2} \) |
| 11 | \( 1 + (2.19 + 15.2i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (-8.96 + 5.76i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-125. + 36.9i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (15.5 + 4.57i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (19.4 - 5.70i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (42.7 - 93.6i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (-16.1 + 18.6i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-14.2 - 16.4i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-144. - 315. i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 - 181.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-281. - 180. i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-7.98 + 5.12i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-216. + 475. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-83.1 + 578. i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (76.1 - 529. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (217. + 63.8i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (981. - 630. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (394. - 454. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (24.7 + 54.2i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (-1.19e3 - 1.38e3i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−12.10920492529023465491685059401, −11.10447885948772204585759041852, −10.24821045579400737574884289263, −9.632993394114190837598368808414, −7.55567712045012649036881396432, −6.66243439809332431448060443672, −5.83503118950861113922788649490, −3.55091370590763560979466448829, −2.49475608049008315602295270897, −1.13122888334364157689835882566,
1.77020848300982861467599239717, 4.29224510169213774138009915588, 5.59303203983321276881187775305, 5.71700299786515992784336873562, 7.43497796465424900061305493617, 8.598727951987532768251647628387, 9.727010008843223862539237558001, 10.35807871790756081920465526962, 11.86589660841257973072194411139, 12.93024867157508770170781707071
