L(s) = 1 | + 0.940i·5-s − 7-s − 5.98i·11-s + 6.59i·13-s − 2.64·17-s − 5.83i·19-s − 2.88·23-s + 4.11·25-s − 3.09i·29-s − 3.52·31-s − 0.940i·35-s + 0.213i·37-s − 1.63·41-s + 7.16i·43-s + 9.32·47-s + ⋯ |
L(s) = 1 | + 0.420i·5-s − 0.377·7-s − 1.80i·11-s + 1.82i·13-s − 0.640·17-s − 1.33i·19-s − 0.601·23-s + 0.823·25-s − 0.575i·29-s − 0.632·31-s − 0.158i·35-s + 0.0351i·37-s − 0.255·41-s + 1.09i·43-s + 1.36·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7337947644\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7337947644\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 0.940iT - 5T^{2} \) |
| 11 | \( 1 + 5.98iT - 11T^{2} \) |
| 13 | \( 1 - 6.59iT - 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 + 5.83iT - 19T^{2} \) |
| 23 | \( 1 + 2.88T + 23T^{2} \) |
| 29 | \( 1 + 3.09iT - 29T^{2} \) |
| 31 | \( 1 + 3.52T + 31T^{2} \) |
| 37 | \( 1 - 0.213iT - 37T^{2} \) |
| 41 | \( 1 + 1.63T + 41T^{2} \) |
| 43 | \( 1 - 7.16iT - 43T^{2} \) |
| 47 | \( 1 - 9.32T + 47T^{2} \) |
| 53 | \( 1 - 7.51iT - 53T^{2} \) |
| 59 | \( 1 - 11.9iT - 59T^{2} \) |
| 61 | \( 1 + 1.48iT - 61T^{2} \) |
| 67 | \( 1 - 13.0iT - 67T^{2} \) |
| 71 | \( 1 - 1.54T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 8.74iT - 83T^{2} \) |
| 89 | \( 1 - 7.50T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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
−8.567842896878346323400646741928, −7.45181477067709864207595398164, −6.84150569438240520102210855824, −6.26821977740282897972971622467, −5.66447143297450662960445025050, −4.53625818901723936804232099519, −3.98415777453166174503844772429, −2.98826942310227399503666454901, −2.38689943519390139789982789313, −1.07729777590360392172838580286,
0.19878271310434394185993081777, 1.54095942128619160383034740016, 2.38602262604711252945416539762, 3.41965437253325142243833808219, 4.14335349680272156224743303049, 5.07049905184906369283746486577, 5.51365065384666571204443218904, 6.46009790278360116593259059024, 7.19596354184515089213198740674, 7.80516159529568812735307595821