L(s) = 1 | + 3.31i·11-s + 2.79·13-s − 4.88i·17-s − 4.35·19-s − 5·25-s + 7·49-s − 8.72i·53-s − 2.72i·59-s − 14.7i·71-s − 17.1·79-s − 1.75i·83-s + 3.27i·89-s − 13.2i·101-s + 17.4·109-s − 20.7i·113-s + ⋯ |
L(s) = 1 | + 1.00i·11-s + 0.775·13-s − 1.18i·17-s − 1.00·19-s − 25-s + 49-s − 1.19i·53-s − 0.355i·59-s − 1.74i·71-s − 1.92·79-s − 0.192i·83-s + 0.346i·89-s − 1.32i·101-s + 1.67·109-s − 1.94i·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.244775186\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.244775186\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.31iT \) |
| 19 | \( 1 + 4.35T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 17 | \( 1 + 4.88iT - 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 8.72iT - 53T^{2} \) |
| 59 | \( 1 + 2.72iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 14.7iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 + 1.75iT - 83T^{2} \) |
| 89 | \( 1 - 3.27iT - 89T^{2} \) |
| 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
−7.61217102575939880045322857028, −7.08183902598457414114434628769, −6.34907938587092690551403175150, −5.65393527705216520274937420835, −4.79583577449806652728662161516, −4.20575996767645532172944864949, −3.37307821788676215199960591400, −2.38119172553117514228671488345, −1.64862461001768516062824284918, −0.32063942211724448659396539156,
1.02000943263608986425193310825, 1.98323766173136007772930153534, 2.95939563736652677395901390828, 3.88122325209651015316477041124, 4.24684519291302496318570751767, 5.49516209442891962852509790612, 5.97843874435572867032659854320, 6.49830505058121752957306682373, 7.43517677280643500312122182905, 8.198987945668227015763917296868