| L(s) = 1 | − 2.33·3-s − 3.10·5-s − 3.54·9-s + 4.69i·11-s + 6.88·13-s + 7.24·15-s − 16.9i·17-s + 26.2·19-s − 25.8·23-s − 15.3·25-s + 29.3·27-s − 42.2i·29-s + 18.3i·31-s − 10.9i·33-s − 49.8i·37-s + ⋯ |
| L(s) = 1 | − 0.778·3-s − 0.620·5-s − 0.393·9-s + 0.426i·11-s + 0.529·13-s + 0.482·15-s − 0.999i·17-s + 1.37·19-s − 1.12·23-s − 0.615·25-s + 1.08·27-s − 1.45i·29-s + 0.592i·31-s − 0.332i·33-s − 1.34i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5395904996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5395904996\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.33T + 9T^{2} \) |
| 5 | \( 1 + 3.10T + 25T^{2} \) |
| 11 | \( 1 - 4.69iT - 121T^{2} \) |
| 13 | \( 1 - 6.88T + 169T^{2} \) |
| 17 | \( 1 + 16.9iT - 289T^{2} \) |
| 19 | \( 1 - 26.2T + 361T^{2} \) |
| 23 | \( 1 + 25.8T + 529T^{2} \) |
| 29 | \( 1 + 42.2iT - 841T^{2} \) |
| 31 | \( 1 - 18.3iT - 961T^{2} \) |
| 37 | \( 1 + 49.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 10.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 13.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 6.97iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 106.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 93.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 89.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 81.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 13.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.15T + 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 88.9iT - 9.40e3T^{2} \) |
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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
−9.590084288662803544830079986583, −8.625677012506568186067113043888, −7.75923162359848413496679194231, −7.14259629546462491701145755960, −6.06047504607692264642830721418, −5.48499412807082860143857360744, −4.51000444653378286676574605825, −3.60795754200473071146284314802, −2.45759670166820050721823155334, −0.889994371960117649030240027877,
0.21932852432097544602386529446, 1.51230437246483833388430405043, 3.11879202931515174523879708105, 3.86934438390273871805338873337, 4.98477513584292767837233192647, 5.81484599940105548883791145560, 6.37597454560381461649351360125, 7.47761611068862464030858932312, 8.200539078210971017424546167442, 8.889004823362544476830938715634
