L(s) = 1 | + (−1.95 + 1.08i)5-s + (−2.37 − 1.16i)7-s + 3.74i·11-s − 0.841·13-s − 3.36i·17-s − 4.55i·19-s + 7.64·23-s + (2.64 − 4.24i)25-s + 1.41i·29-s + 0.979i·31-s + (5.90 − 0.302i)35-s − 2.32i·37-s + 10.3·41-s − 10.8i·43-s − 7.91i·47-s + ⋯ |
L(s) = 1 | + (−0.874 + 0.485i)5-s + (−0.898 − 0.439i)7-s + 1.12i·11-s − 0.233·13-s − 0.814i·17-s − 1.04i·19-s + 1.59·23-s + (0.529 − 0.848i)25-s + 0.262i·29-s + 0.175i·31-s + (0.998 − 0.0511i)35-s − 0.382i·37-s + 1.61·41-s − 1.64i·43-s − 1.15i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9805909349\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9805909349\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.95 - 1.08i)T \) |
| 7 | \( 1 + (2.37 + 1.16i)T \) |
good | 11 | \( 1 - 3.74iT - 11T^{2} \) |
| 13 | \( 1 + 0.841T + 13T^{2} \) |
| 17 | \( 1 + 3.36iT - 17T^{2} \) |
| 19 | \( 1 + 4.55iT - 19T^{2} \) |
| 23 | \( 1 - 7.64T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 0.979iT - 31T^{2} \) |
| 37 | \( 1 + 2.32iT - 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 7.91iT - 47T^{2} \) |
| 53 | \( 1 + 4.35T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 3.74iT - 71T^{2} \) |
| 73 | \( 1 + 8.66T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 3.14iT - 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−9.478583572103335387170784089507, −8.966576200089705652192803158270, −7.63214200438464419421391548917, −7.09633045217886436004226605211, −6.64526391520571500594333465991, −5.16988461506515503596663891636, −4.36654053770684718213690892918, −3.35810631210900332445967868611, −2.49510459271634612090790147147, −0.51467959726166848950940501525,
1.03983222855965268565771391110, 2.86357918863999876592243649597, 3.60396931561888335077244516588, 4.60690983360870288831290366342, 5.74559382293814429853732736315, 6.35237012136414918387350777140, 7.50686110179601435003619986232, 8.224167353027010855832441266652, 8.958456481019972269461630735487, 9.632941673981265481050941640356