L(s) = 1 | + (1.33 − 0.473i)2-s + (0.0574 + 1.73i)3-s + (1.55 − 1.26i)4-s + (−0.461 + 0.266i)5-s + (0.896 + 2.27i)6-s + (0.489 + 2.60i)7-s + (1.46 − 2.41i)8-s + (−2.99 + 0.198i)9-s + (−0.488 + 0.573i)10-s + (2.28 − 3.96i)11-s + (2.27 + 2.61i)12-s − 4.97·13-s + (1.88 + 3.23i)14-s + (−0.487 − 0.783i)15-s + (0.814 − 3.91i)16-s + (2.16 − 3.74i)17-s + ⋯ |
L(s) = 1 | + (0.942 − 0.334i)2-s + (0.0331 + 0.999i)3-s + (0.775 − 0.631i)4-s + (−0.206 + 0.119i)5-s + (0.365 + 0.930i)6-s + (0.184 + 0.982i)7-s + (0.519 − 0.854i)8-s + (−0.997 + 0.0663i)9-s + (−0.154 + 0.181i)10-s + (0.689 − 1.19i)11-s + (0.656 + 0.754i)12-s − 1.38·13-s + (0.503 + 0.864i)14-s + (−0.125 − 0.202i)15-s + (0.203 − 0.979i)16-s + (0.524 − 0.907i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82432 + 0.297232i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82432 + 0.297232i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.33 + 0.473i)T \) |
| 3 | \( 1 + (-0.0574 - 1.73i)T \) |
| 7 | \( 1 + (-0.489 - 2.60i)T \) |
good | 5 | \( 1 + (0.461 - 0.266i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.28 + 3.96i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 + (-2.16 + 3.74i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.921 - 1.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.103 - 0.0596i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.74T + 29T^{2} \) |
| 31 | \( 1 + (1.93 + 1.11i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.02 + 4.05i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 - 1.87iT - 43T^{2} \) |
| 47 | \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.29 - 3.97i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.71 - 3.29i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.07 - 1.86i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 6.05i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.20iT - 71T^{2} \) |
| 73 | \( 1 + (-8.35 - 4.82i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.0228 + 0.0396i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.86iT - 83T^{2} \) |
| 89 | \( 1 + (8.23 + 14.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.18iT - 97T^{2} \) |
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
−12.72861495734043019558915299812, −11.52292165850485823891878327745, −11.38469649335195599261398020121, −9.865700987305922669996992893503, −9.146092891116976551989663158436, −7.57202348543211833616976800296, −5.87832746277666806934946545707, −5.20200419351759905443503252955, −3.81436150626982788833325761948, −2.67710893328147298392608783375,
2.01949235808062537425917585645, 3.84642348580121596924076589633, 5.08328954346981331756380218563, 6.57086185026125423611811337745, 7.33734286560560865912398106832, 8.038046725470234090646682439637, 9.793153791188198859371881620247, 11.19323878322202464542676426602, 12.16573157447482775251917867819, 12.69731195703255133415779959156