L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.160 + 0.909i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s − 0.923·6-s + (1.63 − 1.36i)7-s + (0.5 − 0.866i)8-s + (2.01 − 0.734i)9-s + (0.5 + 0.866i)10-s + (1.53 − 2.65i)11-s + (0.160 − 0.909i)12-s + (3.64 + 1.32i)13-s + (1.06 + 1.84i)14-s + (0.707 + 0.593i)15-s + (0.766 + 0.642i)16-s + (−6.15 + 2.24i)17-s + ⋯ |
L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.0926 + 0.525i)3-s + (−0.469 − 0.171i)4-s + (0.342 − 0.287i)5-s − 0.377·6-s + (0.616 − 0.517i)7-s + (0.176 − 0.306i)8-s + (0.672 − 0.244i)9-s + (0.158 + 0.273i)10-s + (0.461 − 0.799i)11-s + (0.0463 − 0.262i)12-s + (1.01 + 0.367i)13-s + (0.284 + 0.492i)14-s + (0.182 + 0.153i)15-s + (0.191 + 0.160i)16-s + (−1.49 + 0.543i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35925 + 0.629195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35925 + 0.629195i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 37 | \( 1 + (2.88 - 5.35i)T \) |
good | 3 | \( 1 + (-0.160 - 0.909i)T + (-2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (-1.63 + 1.36i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-1.53 + 2.65i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.64 - 1.32i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (6.15 - 2.24i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.665 - 3.77i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-0.436 - 0.756i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.64 + 8.03i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 41 | \( 1 + (-7.73 - 2.81i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 - 5.01T + 43T^{2} \) |
| 47 | \( 1 + (3.62 + 6.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.36 - 7.86i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (8.59 + 7.21i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (11.9 + 4.35i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.31 - 1.10i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0544 - 0.308i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + (-3.39 + 2.84i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (12.0 - 4.38i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (5.18 + 4.35i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.24 - 3.88i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.22096191074398343765849752592, −10.59286822728426511444789478346, −9.457848055555546105203163439135, −8.788996042922655625871854173732, −7.889583927873017519443865971134, −6.65388983081448482799877484582, −5.85845362024801948850856097001, −4.47444807112361048038112048410, −3.83805553837700737030721330397, −1.43947837897745670177523742393,
1.53196611033486691409749194640, 2.53269122436118956525690188157, 4.16471887640298907947326008150, 5.23849438987547569537089656789, 6.68686166528963314298986609155, 7.46536750414521381688537724224, 8.796468702018735865709330124781, 9.282563762946823406298716627170, 10.70118414079811269548107049053, 11.06232355869127382254357403905