| L(s) = 1 | + (−0.959 + 0.281i)3-s + (−1.52 − 0.977i)5-s + (0.485 + 3.37i)7-s + (0.841 − 0.540i)9-s + (−0.800 + 1.75i)11-s + (−0.753 + 5.24i)13-s + (1.73 + 0.509i)15-s + (0.496 + 0.572i)17-s + (−3.10 + 3.58i)19-s + (−1.41 − 3.10i)21-s + (3.36 + 3.42i)23-s + (−0.718 − 1.57i)25-s + (−0.654 + 0.755i)27-s + (−2.79 − 3.22i)29-s + (3.28 + 0.964i)31-s + ⋯ |
| L(s) = 1 | + (−0.553 + 0.162i)3-s + (−0.680 − 0.437i)5-s + (0.183 + 1.27i)7-s + (0.280 − 0.180i)9-s + (−0.241 + 0.528i)11-s + (−0.209 + 1.45i)13-s + (0.448 + 0.131i)15-s + (0.120 + 0.138i)17-s + (−0.711 + 0.821i)19-s + (−0.309 − 0.676i)21-s + (0.700 + 0.713i)23-s + (−0.143 − 0.314i)25-s + (−0.126 + 0.145i)27-s + (−0.518 − 0.598i)29-s + (0.589 + 0.173i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.483559 + 0.579774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.483559 + 0.579774i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-3.36 - 3.42i)T \) |
| good | 5 | \( 1 + (1.52 + 0.977i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.485 - 3.37i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (0.800 - 1.75i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.753 - 5.24i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.496 - 0.572i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (3.10 - 3.58i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (2.79 + 3.22i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-3.28 - 0.964i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (3.09 - 1.98i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (5.92 + 3.80i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-3.52 + 1.03i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + (1.11 + 7.78i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (1.53 - 10.6i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-12.3 - 3.63i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (2.57 + 5.64i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (6.14 + 13.4i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-4.97 + 5.74i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.905 - 6.29i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (8.22 - 5.28i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (10.6 - 3.14i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-6.41 - 4.11i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.98551475254726085901925015019, −11.56562639355952839547751680448, −10.28677866130984377114880538206, −9.205750436261284728679257127508, −8.441215967451043778733574802038, −7.23825790377085536982002319011, −6.04567439626584705271906538444, −4.99426203349899424974806022810, −3.99731986657360150695596211483, −2.05309007266983886379382283212,
0.62038451497262500351657807482, 3.07159653673500045176730343401, 4.32221783044373888637700329096, 5.51495670473448407103630599749, 6.89564829948894252285307849268, 7.52196161508293721276827939069, 8.520863654789971871817204791158, 10.12932026997524136777629290989, 10.82267714131368195089595805620, 11.32670157936830701682652545028
