| L(s) = 1 | + 0.752i·5-s + 3.95·7-s − i·11-s + 0.420i·13-s − 3.06·17-s − 7.03i·19-s + 3.63·23-s + 4.43·25-s − 2.98i·29-s + 2.93·31-s + 2.97i·35-s + 1.20i·37-s − 10.2·41-s − 10.2i·43-s + 11.1·47-s + ⋯ |
| L(s) = 1 | + 0.336i·5-s + 1.49·7-s − 0.301i·11-s + 0.116i·13-s − 0.743·17-s − 1.61i·19-s + 0.757·23-s + 0.886·25-s − 0.555i·29-s + 0.527·31-s + 0.503i·35-s + 0.198i·37-s − 1.59·41-s − 1.56i·43-s + 1.63·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.219487124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.219487124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.752iT - 5T^{2} \) |
| 7 | \( 1 - 3.95T + 7T^{2} \) |
| 13 | \( 1 - 0.420iT - 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 + 7.03iT - 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 + 2.98iT - 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 - 1.20iT - 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 8.38iT - 53T^{2} \) |
| 59 | \( 1 - 1.21iT - 59T^{2} \) |
| 61 | \( 1 + 3.86iT - 61T^{2} \) |
| 67 | \( 1 - 7.79iT - 67T^{2} \) |
| 71 | \( 1 + 6.91T + 71T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 - 6.67T + 79T^{2} \) |
| 83 | \( 1 - 14.8iT - 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 3.22T + 97T^{2} \) |
| 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
−8.761770798619271753199390590506, −7.902198566448936476004968833079, −7.04873122991334646548668032979, −6.58232449239247335370804636876, −5.30622230851197961298688320619, −4.88128048985956198527204309906, −4.02668594409647227140526474623, −2.83137572545730587244916929301, −2.03428027718019002206421124002, −0.78424492399056099691120569795,
1.17179453190661374330963415452, 1.92719080956249082209598265176, 3.11971844565289788841042804735, 4.30542749879157472372460905886, 4.80841596601029455876152062042, 5.55790494882048044267113116851, 6.48518475539730997498213692260, 7.42655319367383039575036793500, 8.008664112169739876827967410791, 8.689667181311369364080684418738
