| L(s) = 1 | − 4.04·5-s + 3.89i·7-s + 0.468i·11-s − 4.89i·13-s + 3.57i·17-s − 1.89·19-s − 8.56·23-s + 11.3·25-s + 2.36·29-s − 7.18i·31-s − 15.7i·35-s − 6.16i·37-s + 5.13i·41-s − 10.5·43-s − 9.37·47-s + ⋯ |
| L(s) = 1 | − 1.80·5-s + 1.47i·7-s + 0.141i·11-s − 1.35i·13-s + 0.867i·17-s − 0.435·19-s − 1.78·23-s + 2.27·25-s + 0.439·29-s − 1.28i·31-s − 2.66i·35-s − 1.01i·37-s + 0.801i·41-s − 1.60·43-s − 1.36·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6807299277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6807299277\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4.04T + 5T^{2} \) |
| 7 | \( 1 - 3.89iT - 7T^{2} \) |
| 11 | \( 1 - 0.468iT - 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 - 3.57iT - 17T^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 23 | \( 1 + 8.56T + 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 + 7.18iT - 31T^{2} \) |
| 37 | \( 1 + 6.16iT - 37T^{2} \) |
| 41 | \( 1 - 5.13iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 + 5.52T + 53T^{2} \) |
| 59 | \( 1 - 6.26iT - 59T^{2} \) |
| 61 | \( 1 + 0.983iT - 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 3.02T + 71T^{2} \) |
| 73 | \( 1 - 4.58T + 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 5.66iT - 83T^{2} \) |
| 89 | \( 1 + 3.17iT - 89T^{2} \) |
| 97 | \( 1 - 0.611T + 97T^{2} \) |
| 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
−8.136500401716218255227990871659, −7.88230616264891481987340351844, −6.73187748041569203422344398553, −5.98755543480053608821769408745, −5.29040131962210554632162387476, −4.37853673370347498449891352686, −3.68101023755143711281551004170, −2.93691507790334697603823523722, −1.96933485590991223777156987876, −0.34174167272872266964324437966,
0.55620694563555298156621512930, 1.78297127471191404526243665226, 3.31523782800664076139442670550, 3.72790498565632422295801730904, 4.52994684306891356131788680280, 4.86831372146002894782184034482, 6.58683602706252690380407770969, 6.76850573858610577270183166802, 7.57525292215001534049592255377, 8.126618924433619675007138279715
