L(s) = 1 | + (0.788 − 1.17i)2-s + (−0.758 − 1.85i)4-s + (0.386 − 2.20i)5-s + (−1.51 − 1.51i)7-s + (−2.77 − 0.568i)8-s + (−2.28 − 2.18i)10-s + 3.92·11-s + (−3.56 + 3.56i)13-s + (−2.97 + 0.585i)14-s + (−2.85 + 2.80i)16-s + (−1.37 + 1.37i)17-s − 4i·19-s + (−4.36 + 0.954i)20-s + (3.09 − 4.60i)22-s + (5.17 − 5.17i)23-s + ⋯ |
L(s) = 1 | + (0.557 − 0.830i)2-s + (−0.379 − 0.925i)4-s + (0.172 − 0.984i)5-s + (−0.573 − 0.573i)7-s + (−0.979 − 0.200i)8-s + (−0.721 − 0.692i)10-s + 1.18·11-s + (−0.988 + 0.988i)13-s + (−0.795 + 0.156i)14-s + (−0.712 + 0.701i)16-s + (−0.333 + 0.333i)17-s − 0.917i·19-s + (−0.976 + 0.213i)20-s + (0.659 − 0.982i)22-s + (1.07 − 1.07i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.488409 - 1.47063i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.488409 - 1.47063i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.788 + 1.17i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.386 + 2.20i)T \) |
good | 7 | \( 1 + (1.51 + 1.51i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 + (3.56 - 3.56i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.37 - 1.37i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-5.17 + 5.17i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 + 7.12iT - 31T^{2} \) |
| 37 | \( 1 + (-3.56 - 3.56i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.75T + 41T^{2} \) |
| 43 | \( 1 + (-5.40 - 5.40i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.54 - 1.54i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.81 + 1.81i)T - 53iT^{2} \) |
| 59 | \( 1 - 3.92iT - 59T^{2} \) |
| 61 | \( 1 - 13.1iT - 61T^{2} \) |
| 67 | \( 1 + (5.40 - 5.40i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 - 5i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 + (6.67 + 6.67i)T + 83iT^{2} \) |
| 89 | \( 1 + 18.4iT - 89T^{2} \) |
| 97 | \( 1 + (10.4 - 10.4i)T - 97iT^{2} \) |
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
−11.32192430557932430580218402646, −10.12203794347477302421201364755, −9.350466017892001200837898406633, −8.764903775233232512261640785489, −6.98943184492741740527330302986, −6.12964899202119178623235455457, −4.62954582586880854403420847725, −4.21778497720116752359392306021, −2.53959667501647327406557116579, −0.941359490234106043160903819761,
2.73802737202922047599124084041, 3.64456491479714458717504118690, 5.15490508790992241656660003092, 6.13058442164067487305827930809, 6.90803457267354733681296451209, 7.74297017322585782194742841474, 9.033496960137500318825100222050, 9.761469056599137957820300162482, 10.99109398920141343812401054524, 12.12990142969961777593741252957