L(s) = 1 | + (1.06 + 0.927i)2-s + 0.662i·3-s + (0.280 + 1.98i)4-s + (−1.31 + 1.81i)5-s + (−0.613 + 0.707i)6-s + (−1.19 − 2.35i)7-s + (−1.53 + 2.37i)8-s + 2.56·9-s + (−3.07 + 0.718i)10-s − 3.09i·11-s + (−1.31 + 0.185i)12-s + 4.66·13-s + (0.905 − 3.63i)14-s + (−1.19 − 0.868i)15-s + (−3.84 + 1.11i)16-s − 2.04·17-s + ⋯ |
L(s) = 1 | + (0.755 + 0.655i)2-s + 0.382i·3-s + (0.140 + 0.990i)4-s + (−0.586 + 0.810i)5-s + (−0.250 + 0.288i)6-s + (−0.453 − 0.891i)7-s + (−0.543 + 0.839i)8-s + 0.853·9-s + (−0.973 + 0.227i)10-s − 0.932i·11-s + (−0.378 + 0.0536i)12-s + 1.29·13-s + (0.242 − 0.970i)14-s + (−0.309 − 0.224i)15-s + (−0.960 + 0.277i)16-s − 0.496·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0152 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0152 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05401 + 1.03805i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05401 + 1.03805i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.06 - 0.927i)T \) |
| 5 | \( 1 + (1.31 - 1.81i)T \) |
| 7 | \( 1 + (1.19 + 2.35i)T \) |
good | 3 | \( 1 - 0.662iT - 3T^{2} \) |
| 11 | \( 1 + 3.09iT - 11T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 + 3.70iT - 37T^{2} \) |
| 41 | \( 1 + 8.48iT - 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 + 0.290iT - 47T^{2} \) |
| 53 | \( 1 - 9.49iT - 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 + 6.45iT - 61T^{2} \) |
| 67 | \( 1 - 2.39T + 67T^{2} \) |
| 71 | \( 1 - 9.65iT - 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 - 1.35iT - 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 2.82iT - 89T^{2} \) |
| 97 | \( 1 - 6.14T + 97T^{2} \) |
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
−13.65795257672862571716112159199, −12.66856066047680389839674930226, −11.26338526965621129497789779085, −10.69265619038189241470160842688, −9.142925198985993268276699138538, −7.70792868395462922787330936263, −6.96170109649813969916091022114, −5.79678763966914802584129745426, −4.01837515623512161543855722977, −3.45910536814225071726263898508,
1.65319984820571781513046393235, 3.60652685828021210322245811121, 4.84691987107443630914521410092, 6.09440818742428017326214753501, 7.43301564518447245073525162626, 8.979828305208503837168184384774, 9.823235704558092799788467492972, 11.28746948012140209214069813918, 12.09795451429030486827263638408, 12.89151057564101011345872257692