L(s) = 1 | + (0.331 − 1.37i)2-s − 2.13i·3-s + (−1.78 − 0.910i)4-s + (−1.94 + 1.10i)5-s + (−2.93 − 0.707i)6-s + (2.35 − 1.19i)7-s + (−1.84 + 2.14i)8-s − 1.56·9-s + (0.874 + 3.03i)10-s − 2.33i·11-s + (−1.94 + 3.80i)12-s − 1.09·13-s + (−0.868 − 3.63i)14-s + (2.35 + 4.15i)15-s + (2.34 + 3.24i)16-s + 4.98·17-s + ⋯ |
L(s) = 1 | + (0.234 − 0.972i)2-s − 1.23i·3-s + (−0.890 − 0.455i)4-s + (−0.869 + 0.493i)5-s + (−1.19 − 0.288i)6-s + (0.891 − 0.453i)7-s + (−0.650 + 0.759i)8-s − 0.520·9-s + (0.276 + 0.961i)10-s − 0.703i·11-s + (−0.561 + 1.09i)12-s − 0.302·13-s + (−0.232 − 0.972i)14-s + (0.608 + 1.07i)15-s + (0.585 + 0.810i)16-s + 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.266654 - 1.01391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.266654 - 1.01391i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.331 + 1.37i)T \) |
| 5 | \( 1 + (1.94 - 1.10i)T \) |
| 7 | \( 1 + (-2.35 + 1.19i)T \) |
good | 3 | \( 1 + 2.13iT - 3T^{2} \) |
| 11 | \( 1 + 2.33iT - 11T^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 - 0.561T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 - 5.49iT - 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 - 9.74iT - 47T^{2} \) |
| 53 | \( 1 - 8.58iT - 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 0.620iT - 61T^{2} \) |
| 67 | \( 1 + 4.71T + 67T^{2} \) |
| 71 | \( 1 + 11.9iT - 71T^{2} \) |
| 73 | \( 1 - 9.96T + 73T^{2} \) |
| 79 | \( 1 - 10.6iT - 79T^{2} \) |
| 83 | \( 1 - 3.86iT - 83T^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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
−12.40229713791391179674715524579, −11.88901931114594916746739457297, −11.05352876766559327582599384825, −9.957304094739802317694088301014, −8.149555024450842805371248101919, −7.71217454923537909615535082135, −6.14494343600400041197244233064, −4.48951877628521748058599165179, −2.99845194580636739954617394539, −1.18660354279074811794868915584,
3.74402375277462123785425278531, 4.70085655387480220777330219126, 5.47931168343339160137457217335, 7.40992093173983531170490050103, 8.242816398514632434876608633415, 9.301170001150430614803781900370, 10.25336626196531126866748852889, 11.81550584240696747465044474005, 12.41718439920853120116773165636, 14.03310420740234949430797191276