| L(s) = 1 | − 0.639·5-s − 3·9-s − 5.62i·11-s + 6.89i·19-s + 4.79i·23-s − 4.59·25-s + 9.59i·31-s + 11.8·37-s − 9.59·41-s + 4.34i·43-s + 1.91·45-s + 2i·47-s − 7·49-s − 13.1·53-s + 3.59i·55-s + ⋯ |
| L(s) = 1 | − 0.285·5-s − 9-s − 1.69i·11-s + 1.58i·19-s + 0.999i·23-s − 0.918·25-s + 1.72i·31-s + 1.95·37-s − 1.49·41-s + 0.662i·43-s + 0.285·45-s + 0.291i·47-s − 49-s − 1.80·53-s + 0.484i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4328718319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4328718319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - 4.79iT \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 0.639T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 5.62iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 6.89iT - 19T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 9.59iT - 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 9.59T + 41T^{2} \) |
| 43 | \( 1 - 4.34iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 8.17iT - 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 + 9.59T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 18.1iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
| 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
−9.766294707960817376040551415133, −8.872629898412281838968013049272, −8.164048207564608910663309529693, −7.69275978182701986686660512677, −6.18859913998721704482359975221, −5.94180079143450307412385630991, −4.87983679306476682090419305836, −3.51543400009347514399929374051, −3.11386387690465826522731801250, −1.46408388628849698568588690651,
0.16841429039001183893974432430, 2.06025100583167401427033202001, 2.91559006285694047322890175204, 4.28012589844985206704149509601, 4.84091796337058703652172376491, 5.97705474554596588089219366629, 6.79944642611029779940907068705, 7.64356471446412348726757580211, 8.327569737620674336898214582151, 9.355351267694286874345858741517
