| L(s) = 1 | + (0.707 − 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s − 2.44i·6-s + (−1.72 + 6.78i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (−9.79 − 16.9i)11-s + (−2.99 − 1.73i)12-s − 2.16i·13-s + (7.08 + 6.91i)14-s + (−2.00 + 3.46i)16-s + (16.9 − 9.81i)17-s + (−2.12 − 3.67i)18-s + (−19.8 − 11.4i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s − 0.408i·6-s + (−0.246 + 0.969i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.890 − 1.54i)11-s + (−0.249 − 0.144i)12-s − 0.166i·13-s + (0.506 + 0.493i)14-s + (−0.125 + 0.216i)16-s + (0.999 − 0.577i)17-s + (−0.117 − 0.204i)18-s + (−1.04 − 0.602i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.367i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9641210003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9641210003\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.72 - 6.78i)T \) |
| good | 11 | \( 1 + (9.79 + 16.9i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 2.16iT - 169T^{2} \) |
| 17 | \( 1 + (-16.9 + 9.81i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (19.8 + 11.4i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (16.2 - 28.1i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 27.6T + 841T^{2} \) |
| 31 | \( 1 + (36.3 - 21.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-30.2 + 52.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 66.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 73.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (52.7 + 30.4i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (46.3 + 80.2i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (24.9 - 14.4i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (72.7 + 42.0i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (1.62 + 2.81i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 38.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-30.5 + 17.6i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (10.7 - 18.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 40.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-42.1 - 24.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 25.2iT - 9.40e3T^{2} \) |
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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
−9.309628225979303634699206202420, −8.435000923697296620868315672645, −7.893462342234344899356611245111, −6.51490941970578760283841568556, −5.70179645632096218516593856146, −4.96839083595693292324507197839, −3.38694863010810945123307701506, −2.97970932420955415592109808808, −1.80443197840378957716205844483, −0.22705496643762405938690980424,
1.86701656922550272315199932731, 3.16691494515101378245800016377, 4.27136712520393500814637687700, 4.70802569206465189560915968402, 6.03492551305154572450567302148, 6.88244266657243224108907415287, 7.78795044254154666057406006682, 8.172828212410886629038873957000, 9.398969588605063021620216246541, 10.30691287055871530664730735911
