L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − i·8-s − 9-s + 4·11-s − i·12-s − i·13-s + 16-s − 6i·17-s − i·18-s − 4·19-s + 4i·22-s − 8i·23-s + 24-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + 1.20·11-s − 0.288i·12-s − 0.277i·13-s + 0.250·16-s − 1.45i·17-s − 0.235i·18-s − 0.917·19-s + 0.852i·22-s − 1.66i·23-s + 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.120125027\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.120125027\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−9.154252155014664858822159767044, −8.487864019511558050260952450192, −7.43308946362867899085930968108, −6.79355452885207774902165474077, −5.94652991610837050895795060529, −5.12496116293216192488918629710, −4.26387253987100039449342818826, −3.55723636751264796696742918548, −2.21065775232864566041423645635, −0.41220225760447435717141390099,
1.41712201181598111067079437900, 1.98132603108891222983698033928, 3.50815152837742043223745463572, 3.95441788940311451170544654144, 5.21510627854161874037652402094, 6.12404854671603562682731776745, 6.82520505257646537146941498508, 7.79524108488206212661689876148, 8.589440121104046330999730805296, 9.236169835844586346852186900313