L(s) = 1 | + (−1 − i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (−1.22 + 1.87i)5-s − 2.44·6-s + (−1.87 − 1.87i)7-s + (2 − 2i)8-s − 2.99i·9-s + (3.09 − 0.646i)10-s − 3.58·11-s + (2.44 + 2.44i)12-s + 3.74i·14-s + (0.791 + 3.79i)15-s − 4·16-s + (−5.54 − 5.54i)17-s + (−2.99 + 2.99i)18-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + i·4-s + (−0.547 + 0.836i)5-s − 0.999·6-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s − 0.999i·9-s + (0.978 − 0.204i)10-s − 1.08·11-s + (0.707 + 0.707i)12-s + 0.999i·14-s + (0.204 + 0.978i)15-s − 16-s + (−1.34 − 1.34i)17-s + (−0.707 + 0.707i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00695475 + 0.533609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00695475 + 0.533609i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 + (1.22 - 1.87i)T \) |
| 7 | \( 1 + (1.87 + 1.87i)T \) |
good | 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (5.54 + 5.54i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.15iT - 19T^{2} \) |
| 23 | \( 1 + (-4.58 + 4.58i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 9.93T + 31T^{2} \) |
| 37 | \( 1 + (-8.58 - 8.58i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.74T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 6.41T + 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 - 2.44iT - 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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
−10.84363622847354788067790097923, −9.805181394628411539194976695755, −8.986032627239052376229110668861, −7.933230412967241165686779023264, −7.18644078486464344237520478801, −6.66390252819599938138418415904, −4.37270891342791932872715023595, −3.10661032333012854242222008376, −2.51318630973689709965962594106, −0.37127806488961314713310430982,
2.18583346207856863540418543151, 3.84360415510670066324111612586, 5.05673679003158300997723780958, 5.87261401342179626859542133737, 7.38839363562173544864637339894, 8.153717935304558114310460859319, 9.029679169271077077027858669533, 9.356482289916967437592808656861, 10.57441275433354245271017673343, 11.23353207636320723733216057138