| L(s) = 1 | + (0.0345 − 0.240i)2-s + (0.654 + 0.755i)3-s + (1.86 + 0.546i)4-s + (0.0517 + 0.0332i)5-s + (0.204 − 0.131i)6-s + (−0.587 + 4.08i)7-s + (0.397 − 0.870i)8-s + (−0.142 + 0.989i)9-s + (0.00977 − 0.0112i)10-s + (−3.49 − 2.24i)11-s + (0.806 + 1.76i)12-s + (−2.09 − 4.59i)13-s + (0.962 + 0.282i)14-s + (0.00874 + 0.0608i)15-s + (3.07 + 1.97i)16-s + (6.79 − 1.99i)17-s + ⋯ |
| L(s) = 1 | + (0.0244 − 0.169i)2-s + (0.378 + 0.436i)3-s + (0.931 + 0.273i)4-s + (0.0231 + 0.0148i)5-s + (0.0833 − 0.0535i)6-s + (−0.222 + 1.54i)7-s + (0.140 − 0.307i)8-s + (−0.0474 + 0.329i)9-s + (0.00309 − 0.00356i)10-s + (−1.05 − 0.676i)11-s + (0.232 + 0.509i)12-s + (−0.581 − 1.27i)13-s + (0.257 + 0.0755i)14-s + (0.00225 + 0.0157i)15-s + (0.767 + 0.493i)16-s + (1.64 − 0.483i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47056 + 0.449464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47056 + 0.449464i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 67 | \( 1 + (4.43 - 6.88i)T \) |
| good | 2 | \( 1 + (-0.0345 + 0.240i)T + (-1.91 - 0.563i)T^{2} \) |
| 5 | \( 1 + (-0.0517 - 0.0332i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.587 - 4.08i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (3.49 + 2.24i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.09 + 4.59i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-6.79 + 1.99i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.543 - 3.78i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (2.50 + 2.89i)T + (-3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 - 2.91T + 29T^{2} \) |
| 31 | \( 1 + (-1.46 + 3.21i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 - 7.30T + 37T^{2} \) |
| 41 | \( 1 + (10.6 - 3.12i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (2.12 - 0.622i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (7.24 + 8.35i)T + (-6.68 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.42 - 0.419i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (4.22 - 9.25i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (2.91 - 1.87i)T + (25.3 - 55.4i)T^{2} \) |
| 71 | \( 1 + (5.19 + 1.52i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-1.60 + 1.03i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.180 - 0.395i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-14.5 - 9.34i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (3.58 - 4.14i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 - 7.44T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.24379571011478538938616835654, −11.85096621743256744744789377011, −10.36404435108899776304566191691, −9.915653031024319861397684401045, −8.246378406766750196224965248331, −7.892381504842829553189209019419, −6.06269653257033128128631885336, −5.33150529425758099614554515715, −3.18877117321255296800972752757, −2.54480118293882426398154875155,
1.65249769809901830571277764665, 3.27638771987497894829939174662, 4.89779114107336065154193740092, 6.45024497548546012125901025425, 7.37544011282170584501440867419, 7.79324226232476479512095820877, 9.690201961615324372577353892608, 10.29752129204869745649450232836, 11.40593070445072800766370461017, 12.36758173631616550409632617960
