L(s) = 1 | + (−0.707 − 2.12i)5-s + (−2 − 2i)7-s − 2.82i·11-s + (−3 + 3i)13-s + (−2.82 + 2.82i)17-s + 8i·19-s + (−2.82 − 2.82i)23-s + (−3.99 + 3i)25-s + 1.41·29-s − 4·31-s + (−2.82 + 5.65i)35-s + (−3 − 3i)37-s − 9.89i·41-s + i·49-s + (−2.82 − 2.82i)53-s + ⋯ |
L(s) = 1 | + (−0.316 − 0.948i)5-s + (−0.755 − 0.755i)7-s − 0.852i·11-s + (−0.832 + 0.832i)13-s + (−0.685 + 0.685i)17-s + 1.83i·19-s + (−0.589 − 0.589i)23-s + (−0.799 + 0.600i)25-s + 0.262·29-s − 0.718·31-s + (−0.478 + 0.956i)35-s + (−0.493 − 0.493i)37-s − 1.54i·41-s + 0.142i·49-s + (−0.388 − 0.388i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0128419 + 0.320910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0128419 + 0.320910i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 + 2.12i)T \) |
good | 7 | \( 1 + (2 + 2i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.82 - 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (2.82 + 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (-4 - 4i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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
−9.993503964636198224030512546948, −9.053892995357185327056937728037, −8.327695438871047591496395092994, −7.40825693350748097855272874510, −6.40737718812415350190111408752, −5.48589798083742353203024673395, −4.21356399565083671514379527282, −3.64268188364035595652042383880, −1.84347149441485277805172706486, −0.15556259102759866391812967000,
2.44735317854314536331101603992, 3.03617743366233901509756974319, 4.48317662742546164836781586482, 5.50000292485725620623954091062, 6.70808549194727805996555403117, 7.14667016751687632627091982644, 8.195184285276335669390959872640, 9.436084026958537183416506476625, 9.800312783855903857101266230328, 10.90261471581817699983221699424