L(s) = 1 | + 4.84i·3-s + 6.85·7-s − 14.4·9-s − 9.59·11-s − 10.0i·13-s + 27.8·17-s + (17.5 + 7.24i)19-s + 33.2i·21-s − 8.14·23-s − 26.3i·27-s + 17.8i·29-s + 20.3i·31-s − 46.4i·33-s + 65.1i·37-s + 48.8·39-s + ⋯ |
L(s) = 1 | + 1.61i·3-s + 0.979·7-s − 1.60·9-s − 0.872·11-s − 0.775i·13-s + 1.63·17-s + (0.924 + 0.381i)19-s + 1.58i·21-s − 0.354·23-s − 0.977i·27-s + 0.614i·29-s + 0.657i·31-s − 1.40i·33-s + 1.75i·37-s + 1.25·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.855153090\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.855153090\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-17.5 - 7.24i)T \) |
good | 3 | \( 1 - 4.84iT - 9T^{2} \) |
| 7 | \( 1 - 6.85T + 49T^{2} \) |
| 11 | \( 1 + 9.59T + 121T^{2} \) |
| 13 | \( 1 + 10.0iT - 169T^{2} \) |
| 17 | \( 1 - 27.8T + 289T^{2} \) |
| 23 | \( 1 + 8.14T + 529T^{2} \) |
| 29 | \( 1 - 17.8iT - 841T^{2} \) |
| 31 | \( 1 - 20.3iT - 961T^{2} \) |
| 37 | \( 1 - 65.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 16.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 60.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.60T + 2.20e3T^{2} \) |
| 53 | \( 1 - 14.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 0.442iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 65.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 103. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 88.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 86.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 47.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 58.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 165. iT - 9.40e3T^{2} \) |
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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
−9.719694860522175348061163764169, −8.405733063854449231661067039260, −8.191404626892753345581397626464, −7.18543028170965127817895542494, −5.60467062273909902155714084296, −5.31691458566626819317483956677, −4.61935619451617860788481337106, −3.49901908091427177218416587783, −2.94342867846776410570463770994, −1.28416899019061988348917526701,
0.50053321983952689995644603285, 1.56454232030703119870113059837, 2.28598419355419452562293220478, 3.42734611966638838629022538638, 4.83557507148140278166577479828, 5.58902782007957728304221917991, 6.37212700705566723128022740161, 7.44966690555506094814544440209, 7.68899662967546364130059263417, 8.315897416735724958425433321841