| L(s) = 1 | + 9.32i·7-s + 19.3i·11-s + 19.6i·13-s + 4.70·17-s + 29.9·19-s − 9.89·23-s + 24.5i·29-s − 12.6·31-s − 37.9i·37-s − 73.0i·41-s + 33.3i·43-s − 4.70·47-s − 37.9·49-s − 73.0·53-s − 24.5i·59-s + ⋯ |
| L(s) = 1 | + 1.33i·7-s + 1.75i·11-s + 1.51i·13-s + 0.276·17-s + 1.57·19-s − 0.430·23-s + 0.844i·29-s − 0.408·31-s − 1.02i·37-s − 1.78i·41-s + 0.774i·43-s − 0.100·47-s − 0.774·49-s − 1.37·53-s − 0.415i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.629143880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.629143880\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 9.32iT - 49T^{2} \) |
| 11 | \( 1 - 19.3iT - 121T^{2} \) |
| 13 | \( 1 - 19.6iT - 169T^{2} \) |
| 17 | \( 1 - 4.70T + 289T^{2} \) |
| 19 | \( 1 - 29.9T + 361T^{2} \) |
| 23 | \( 1 + 9.89T + 529T^{2} \) |
| 29 | \( 1 - 24.5iT - 841T^{2} \) |
| 31 | \( 1 + 12.6T + 961T^{2} \) |
| 37 | \( 1 + 37.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 73.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 33.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 4.70T + 2.20e3T^{2} \) |
| 53 | \( 1 + 73.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 53.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 101. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 111. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 10.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 44.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 38.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 183. iT - 9.40e3T^{2} \) |
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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
−9.278810169749026279064416965197, −8.972430703870593757163162012863, −7.68201447742772994789051983790, −7.15325570354355025945862037680, −6.22480603049452530028170663436, −5.29019715827572746410237999589, −4.64543194468615748441970552798, −3.54062841426427555020816070425, −2.29985748461777361992104982462, −1.65645120577010258914505224167,
0.46223024212675292003053269706, 1.17666742806195462335996364297, 3.12430249960742555219878264050, 3.39970027687078991890028145985, 4.63807002462408084320694607639, 5.62454061454782018625962281869, 6.22664508586248572768831833477, 7.40880136780897943917719302828, 7.895884237345929429389600740929, 8.570766874108309129992608756689
