| L(s) = 1 | + 1.59·5-s − 1.16i·7-s + 1.98i·11-s − 0.164i·13-s + 3.57i·17-s − 3.16·19-s − 1.21·23-s − 2.44·25-s + 4.98·29-s + 6.64i·31-s − 1.85i·35-s + 1.10i·37-s + 2.02i·41-s − 8.34·43-s + 2.21·47-s + ⋯ |
| L(s) = 1 | + 0.714·5-s − 0.440i·7-s + 0.597i·11-s − 0.0455i·13-s + 0.867i·17-s − 0.725·19-s − 0.253·23-s − 0.489·25-s + 0.925·29-s + 1.19i·31-s − 0.314i·35-s + 0.181i·37-s + 0.316i·41-s − 1.27·43-s + 0.323·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.378229445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378229445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 + 1.16iT - 7T^{2} \) |
| 11 | \( 1 - 1.98iT - 11T^{2} \) |
| 13 | \( 1 + 0.164iT - 13T^{2} \) |
| 17 | \( 1 - 3.57iT - 17T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 + 1.21T + 23T^{2} \) |
| 29 | \( 1 - 4.98T + 29T^{2} \) |
| 31 | \( 1 - 6.64iT - 31T^{2} \) |
| 37 | \( 1 - 1.10iT - 37T^{2} \) |
| 41 | \( 1 - 2.02iT - 41T^{2} \) |
| 43 | \( 1 + 8.34T + 43T^{2} \) |
| 47 | \( 1 - 2.21T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 + 0.888iT - 59T^{2} \) |
| 61 | \( 1 - 9.74iT - 61T^{2} \) |
| 67 | \( 1 + 3.98T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 0.879T + 73T^{2} \) |
| 79 | \( 1 - 2.91iT - 79T^{2} \) |
| 83 | \( 1 - 13.9iT - 83T^{2} \) |
| 89 | \( 1 - 8.41iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
| 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
−8.441377464410973417677239118488, −7.72760298279545985920225046281, −6.85806825219414922724844658592, −6.33643896286388679956341070963, −5.58326040371530560925262299881, −4.70615971424586645647767316056, −4.05682581731179065187685018704, −3.06462427534613053657796300906, −2.06370955647408792329272884032, −1.29989682892482458509046997435,
0.34649463973286440559351091050, 1.73216768117422008690352035312, 2.50914172179415519412397505176, 3.35030017190523906693991889160, 4.39305164132004988138131393838, 5.13576110687953476188315977645, 6.00531270313955161259143429986, 6.30120772934248119315574279908, 7.31232459989364851204019832708, 8.044672619183861097440618611813
