| L(s) = 1 | − 2.16i·2-s − 0.674·4-s − 7.64·5-s + 6.05i·7-s − 7.18i·8-s + 16.5i·10-s − 3.21i·13-s + 13.0·14-s − 18.2·16-s − 6.22i·17-s + 19.4i·19-s + 5.15·20-s − 3.47·23-s + 33.3·25-s − 6.94·26-s + ⋯ |
| L(s) = 1 | − 1.08i·2-s − 0.168·4-s − 1.52·5-s + 0.865i·7-s − 0.898i·8-s + 1.65i·10-s − 0.247i·13-s + 0.935·14-s − 1.14·16-s − 0.366i·17-s + 1.02i·19-s + 0.257·20-s − 0.150·23-s + 1.33·25-s − 0.267·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.343697992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343697992\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.16iT - 4T^{2} \) |
| 5 | \( 1 + 7.64T + 25T^{2} \) |
| 7 | \( 1 - 6.05iT - 49T^{2} \) |
| 13 | \( 1 + 3.21iT - 169T^{2} \) |
| 17 | \( 1 + 6.22iT - 289T^{2} \) |
| 19 | \( 1 - 19.4iT - 361T^{2} \) |
| 23 | \( 1 + 3.47T + 529T^{2} \) |
| 29 | \( 1 - 36.1iT - 841T^{2} \) |
| 31 | \( 1 + 3.28T + 961T^{2} \) |
| 37 | \( 1 - 63.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 31.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 34.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 16.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 63.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 110. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 96.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 44.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 63.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 14.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 88.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 51.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 31.5T + 9.40e3T^{2} \) |
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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
−9.688878156310031610478906711396, −8.795385680374545122626026763657, −7.964944875883742545080037611539, −7.21450217788781463374328893331, −6.16010527513874552904466924752, −4.94379919969931563079926837969, −3.87577245468596880309141544053, −3.24271106754642211314557813473, −2.18202574299781208474776647588, −0.74738029282685022709065405341,
0.64448955758652637272623950640, 2.59270381909725060813726729110, 3.99335773837063110737406188996, 4.48328360197024451191855851497, 5.70854692499777958055323916800, 6.76024233190904572991498258051, 7.29651146741393906313698509485, 7.954898904933616680233895006229, 8.536215139379695089783386815058, 9.651067189901586620691137101828
