| L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 0.353·5-s − 0.790i·7-s − 2.82i·8-s − 0.499i·10-s − 0.209i·13-s + 1.11·14-s + 4.00·16-s + 14.9i·17-s − 2.16i·19-s + 0.706·20-s + 24.1·23-s − 24.8·25-s + 0.295·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.0706·5-s − 0.112i·7-s − 0.353i·8-s − 0.0499i·10-s − 0.0160i·13-s + 0.0798·14-s + 0.250·16-s + 0.880i·17-s − 0.114i·19-s + 0.0353·20-s + 1.04·23-s − 0.995·25-s + 0.0113·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.576700663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576700663\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 0.353T + 25T^{2} \) |
| 7 | \( 1 + 0.790iT - 49T^{2} \) |
| 13 | \( 1 + 0.209iT - 169T^{2} \) |
| 17 | \( 1 - 14.9iT - 289T^{2} \) |
| 19 | \( 1 + 2.16iT - 361T^{2} \) |
| 23 | \( 1 - 24.1T + 529T^{2} \) |
| 29 | \( 1 + 17.8iT - 841T^{2} \) |
| 31 | \( 1 - 33.8T + 961T^{2} \) |
| 37 | \( 1 + 60.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 53.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 15.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 61.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 58.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 51.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 40.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 72.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 95.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 113. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 147. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 19.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 74.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 101.T + 9.40e3T^{2} \) |
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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
−8.868827193886764028453770084665, −8.350266529213105653905167962677, −7.47515733565209415297654531837, −6.84152893642673039159756927123, −5.97595384669160082854609543424, −5.28341705403324328287242047119, −4.29509022695132092049524701904, −3.56028659025835017853070353793, −2.27964192502332440391882546189, −0.904048258980559411439481648617,
0.49853015206727218683293983309, 1.69368553444763190637095653824, 2.77996372772790384471753527525, 3.54224516139014092755365227138, 4.61952222583194964429670301087, 5.25735949899465304245968652242, 6.28008716631309895614960627908, 7.19940665933889476710645758801, 7.976768275835145683943218762349, 8.900439639444821541880283100807
