| L(s) = 1 | − 2i·3-s − i·7-s − 9-s + 11-s − 4i·13-s − 6·19-s − 2·21-s − 3i·23-s − 4i·27-s + 3·29-s − 2i·33-s + 9i·37-s − 8·39-s + 2·41-s − 9i·43-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s − 0.377i·7-s − 0.333·9-s + 0.301·11-s − 1.10i·13-s − 1.37·19-s − 0.436·21-s − 0.625i·23-s − 0.769i·27-s + 0.557·29-s − 0.348i·33-s + 1.47i·37-s − 1.28·39-s + 0.312·41-s − 1.37i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.322561434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322561434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 9iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 9iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + iT - 67T^{2} \) |
| 71 | \( 1 + 7T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 9T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.017677189013623707133571075219, −8.204675737945985483454912992224, −7.65984502706950344450940204942, −6.66561719375102986601028406148, −6.28598407300574551655310136465, −5.07063496362356523054172266486, −4.03881938396490623702544436908, −2.82801833982336992243392563522, −1.73907920124756321652326457018, −0.53235295144991596471459908194,
1.74823676947384655770563592336, 3.03642885803833412542292928486, 4.22386622710022987780700289512, 4.52035633040959335322798926595, 5.73249939628901902563791392083, 6.50816979297899993781651137106, 7.51063017902087504211867564221, 8.595105853823862202606009325874, 9.272173387758320515469722329583, 9.697088823818481711701633072257
