L(s) = 1 | + (−1.13 + 2.77i)3-s + (6.28 + 6.28i)5-s + 1.64i·7-s + (−6.43 − 6.29i)9-s + (4.75 + 4.75i)11-s + (−9.35 − 9.35i)13-s + (−24.5 + 10.3i)15-s + 11.4i·17-s + (8.58 + 8.58i)19-s + (−4.57 − 1.86i)21-s − 16.2·23-s + 54.0i·25-s + (24.7 − 10.7i)27-s + (−10.7 + 10.7i)29-s − 6.35·31-s + ⋯ |
L(s) = 1 | + (−0.377 + 0.926i)3-s + (1.25 + 1.25i)5-s + 0.235i·7-s + (−0.715 − 0.699i)9-s + (0.432 + 0.432i)11-s + (−0.719 − 0.719i)13-s + (−1.63 + 0.689i)15-s + 0.675i·17-s + (0.451 + 0.451i)19-s + (−0.217 − 0.0887i)21-s − 0.706·23-s + 2.16i·25-s + (0.917 − 0.398i)27-s + (−0.370 + 0.370i)29-s − 0.204·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.806900 + 1.25734i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.806900 + 1.25734i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.13 - 2.77i)T \) |
good | 5 | \( 1 + (-6.28 - 6.28i)T + 25iT^{2} \) |
| 7 | \( 1 - 1.64iT - 49T^{2} \) |
| 11 | \( 1 + (-4.75 - 4.75i)T + 121iT^{2} \) |
| 13 | \( 1 + (9.35 + 9.35i)T + 169iT^{2} \) |
| 17 | \( 1 - 11.4iT - 289T^{2} \) |
| 19 | \( 1 + (-8.58 - 8.58i)T + 361iT^{2} \) |
| 23 | \( 1 + 16.2T + 529T^{2} \) |
| 29 | \( 1 + (10.7 - 10.7i)T - 841iT^{2} \) |
| 31 | \( 1 + 6.35T + 961T^{2} \) |
| 37 | \( 1 + (-27.2 + 27.2i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 1.98T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-19.4 + 19.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 74.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-4.00 - 4.00i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-27.9 - 27.9i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-39.2 - 39.2i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-68.6 - 68.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 40.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 17.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-75.1 + 75.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 78.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 38.8T + 9.40e3T^{2} \) |
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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
−12.51274798771272904676175606371, −11.42461430594443546018159618636, −10.29411976675848176471144865381, −10.05197942352361360281944861376, −8.981102829958163908851309317914, −7.29490887946313932298775813869, −6.09911554270847758220018262699, −5.40473440215441547316529739911, −3.71255868839908112490763650475, −2.34632110917814461012773088662,
0.947615677416685273639975781063, 2.26395594157729484041766228028, 4.68777704979338337160627900663, 5.67832602391305649720587915049, 6.63539223168026691223367692186, 7.899052966255246042914415121524, 9.087397629123188104582817602878, 9.771656937236842272868147856964, 11.27693858541292827261598150011, 12.15509820382779710369768253624