L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s + 2.44i·6-s + (−6.72 − 1.94i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (0.263 − 0.455i)11-s + (−1.73 − 2.99i)12-s + 4.22·13-s + (9.61 − 2.37i)14-s + (−2.00 − 3.46i)16-s + (−16.6 + 28.8i)17-s + (3.67 + 2.12i)18-s + (−2.75 + 1.58i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.288 − 0.5i)3-s + (0.249 − 0.433i)4-s + 0.408i·6-s + (−0.960 − 0.277i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.0239 − 0.0414i)11-s + (−0.144 − 0.249i)12-s + 0.324·13-s + (0.686 − 0.169i)14-s + (−0.125 − 0.216i)16-s + (−0.980 + 1.69i)17-s + (0.204 + 0.117i)18-s + (−0.144 + 0.0836i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.200237416\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.200237416\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.72 + 1.94i)T \) |
good | 11 | \( 1 + (-0.263 + 0.455i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 4.22T + 169T^{2} \) |
| 17 | \( 1 + (16.6 - 28.8i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (2.75 - 1.58i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-9.57 + 5.52i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 56.1T + 841T^{2} \) |
| 31 | \( 1 + (1.63 + 0.945i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-8.44 + 4.87i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 4.07iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 46.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (31.6 + 54.7i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-40.2 - 23.2i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-43.7 - 25.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-89.4 + 51.6i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-7.56 - 4.36i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 29.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (8.36 - 14.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (66.1 + 114. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 12.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-59.1 + 34.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 149.T + 9.40e3T^{2} \) |
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
−9.740309525595513668476077863844, −8.590689514228045056110766984125, −8.406410336692393575419310769258, −7.15549573364430169707707107666, −6.51178344643783851420330506661, −5.94951795249355541508378826341, −4.45468081238490173273657933870, −3.33726138597561975020861444665, −2.14570233985310514275629583327, −0.821037404905924452025408509448,
0.62371555882698360860081706496, 2.42420902551548187126564657131, 3.10038456120361556919242659722, 4.23292659518725451595147388976, 5.28198590342299365821860182270, 6.55417140061673173355663284243, 7.13378061247614145310323745674, 8.388124747845347475781647442146, 8.961279972318675978195691379094, 9.676236781125728639606741347358