| L(s) = 1 | + 2.14i·3-s + 9.06i·7-s + 4.41·9-s − 4.28i·11-s + 9.41·13-s − 18·17-s + 36.2i·19-s − 19.4·21-s − 22.9i·23-s + 28.7i·27-s − 44.8·29-s + 35.2i·31-s + 9.16·33-s − 6.58·37-s + 20.1i·39-s + ⋯ |
| L(s) = 1 | + 0.713i·3-s + 1.29i·7-s + 0.490·9-s − 0.389i·11-s + 0.724·13-s − 1.05·17-s + 1.90i·19-s − 0.924·21-s − 0.996i·23-s + 1.06i·27-s − 1.54·29-s + 1.13i·31-s + 0.277·33-s − 0.177·37-s + 0.516i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.759495 + 1.31548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.759495 + 1.31548i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.14iT - 9T^{2} \) |
| 7 | \( 1 - 9.06iT - 49T^{2} \) |
| 11 | \( 1 + 4.28iT - 121T^{2} \) |
| 13 | \( 1 - 9.41T + 169T^{2} \) |
| 17 | \( 1 + 18T + 289T^{2} \) |
| 19 | \( 1 - 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 22.9iT - 529T^{2} \) |
| 29 | \( 1 + 44.8T + 841T^{2} \) |
| 31 | \( 1 - 35.2iT - 961T^{2} \) |
| 37 | \( 1 + 6.58T + 1.36e3T^{2} \) |
| 41 | \( 1 - 52.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 28.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 90.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 52.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 17.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 50.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 33.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 20.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 91.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 42.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 22.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 47.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 160.T + 9.40e3T^{2} \) |
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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
−11.15000149856074500658154624326, −10.54752093613568379568362419059, −9.405106244069913301884222809382, −8.834433228648131250019721566610, −7.84996861427044406594145598653, −6.38604832355003618320412593169, −5.61836130131158733517337100905, −4.43567374065122520394415311829, −3.36740599326773650731463560983, −1.85041083219938140380351979676,
0.67313751608410153172722189414, 2.04259590323300974719653698673, 3.78301534386117197411365360535, 4.69902783616073130992015576810, 6.24950296608100761148437604098, 7.18164334360630482821429264944, 7.57910497345413696887346683788, 8.960453388153512301876502061978, 9.834773817080084592803463004735, 10.98054683495287956491086378383
