| L(s) = 1 | + (0.658 − 0.156i)2-s + (1.15 − 1.28i)3-s + (−1.37 + 0.692i)4-s + (0.563 + 0.756i)5-s + (0.560 − 1.02i)6-s + (1.22 − 0.802i)7-s + (−1.83 + 1.54i)8-s + (−0.324 − 2.98i)9-s + (0.488 + 0.410i)10-s + (−6.07 − 0.710i)11-s + (−0.701 + 2.57i)12-s + (1.50 + 5.02i)13-s + (0.678 − 0.718i)14-s + (1.62 + 0.148i)15-s + (0.873 − 1.17i)16-s + (−0.00536 + 0.0304i)17-s + ⋯ |
| L(s) = 1 | + (0.465 − 0.110i)2-s + (0.667 − 0.744i)3-s + (−0.689 + 0.346i)4-s + (0.251 + 0.338i)5-s + (0.228 − 0.420i)6-s + (0.461 − 0.303i)7-s + (−0.649 + 0.544i)8-s + (−0.108 − 0.994i)9-s + (0.154 + 0.129i)10-s + (−1.83 − 0.214i)11-s + (−0.202 + 0.744i)12-s + (0.417 + 1.39i)13-s + (0.181 − 0.192i)14-s + (0.420 + 0.0384i)15-s + (0.218 − 0.293i)16-s + (−0.00130 + 0.00737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19539 - 0.244694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19539 - 0.244694i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.15 + 1.28i)T \) |
| good | 2 | \( 1 + (-0.658 + 0.156i)T + (1.78 - 0.897i)T^{2} \) |
| 5 | \( 1 + (-0.563 - 0.756i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (-1.22 + 0.802i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (6.07 + 0.710i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (-1.50 - 5.02i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (0.00536 - 0.0304i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.634 - 3.60i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (0.0873 + 0.0574i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (0.351 + 0.372i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.475 + 8.15i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (-5.42 + 1.97i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-4.26 - 1.01i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (-0.545 - 1.26i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (0.326 + 5.60i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (4.89 - 8.47i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.61 + 0.655i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (-1.53 - 0.770i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (1.87 - 1.99i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-1.66 - 1.39i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (6.38 - 5.35i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (12.2 - 2.90i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (-17.3 + 4.11i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (10.6 - 8.96i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (4.18 - 5.62i)T + (-27.8 - 92.9i)T^{2} \) |
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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
−14.00645944850592540322838693669, −13.41895805406226211867568241902, −12.49963700596156106032882369239, −11.25800795679910519517895088749, −9.718590439420850352264976471949, −8.416089480787785572411142942377, −7.60186514693830816972114120059, −5.92742095323185510942204254356, −4.21565118652582464280513870094, −2.56756167427398023149752262738,
3.04333113009479449928051231715, 4.89174771744846840816287325156, 5.43648157697785423328567007687, 7.892688110967578603507253835829, 8.824638550275261887397470863925, 9.993715623826450146824463552063, 10.85247850210870889817348585466, 12.91360949207815288393212511474, 13.28683811747923075917150856108, 14.51404923978998856479479786785
