L(s) = 1 | + (0.532 − 1.19i)2-s + (−0.567 + 2.67i)3-s + (0.190 + 0.211i)4-s + (0.406 + 0.0427i)5-s + (2.89 + 2.10i)6-s + (−1.52 − 2.15i)7-s + (2.84 − 0.924i)8-s + (−4.07 − 1.81i)9-s + (0.268 − 0.464i)10-s + (−2.14 − 2.53i)11-s + (−0.671 + 0.387i)12-s + (−0.864 + 0.628i)13-s + (−3.39 + 0.680i)14-s + (−0.345 + 1.06i)15-s + (0.350 − 3.33i)16-s + (5.03 − 2.24i)17-s + ⋯ |
L(s) = 1 | + (0.376 − 0.846i)2-s + (−0.327 + 1.54i)3-s + (0.0950 + 0.105i)4-s + (0.182 + 0.0191i)5-s + (1.18 + 0.858i)6-s + (−0.578 − 0.815i)7-s + (1.00 − 0.326i)8-s + (−1.35 − 0.604i)9-s + (0.0847 − 0.146i)10-s + (−0.646 − 0.763i)11-s + (−0.193 + 0.111i)12-s + (−0.239 + 0.174i)13-s + (−0.908 + 0.181i)14-s + (−0.0891 + 0.274i)15-s + (0.0875 − 0.833i)16-s + (1.22 − 0.543i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06977 + 0.0824229i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06977 + 0.0824229i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (1.52 + 2.15i)T \) |
| 11 | \( 1 + (2.14 + 2.53i)T \) |
good | 2 | \( 1 + (-0.532 + 1.19i)T + (-1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (0.567 - 2.67i)T + (-2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.406 - 0.0427i)T + (4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (0.864 - 0.628i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.03 + 2.24i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (3.06 - 3.39i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.37 - 2.38i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.06 - 0.345i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.74 - 0.709i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-6.56 + 1.39i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (2.68 + 8.25i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.85iT - 43T^{2} \) |
| 47 | \( 1 + (-1.94 - 1.74i)T + (4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-1.40 - 13.3i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-1.55 + 1.40i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.746 - 7.10i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-4.12 + 7.14i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.08 - 1.51i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (6.49 + 7.21i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (0.844 - 1.89i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (2.94 + 2.14i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.68 + 1.54i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.81 - 10.7i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−14.39540733203928122410225498772, −13.36826624275925792224011174785, −12.14232346812583291051655052595, −10.95088398545048378641080128564, −10.37204161438622889020136093839, −9.498052908048693080770125053911, −7.61820558453787618832682437876, −5.66405504093169169718819252298, −4.18370003998941765193725083991, −3.22334683058074935221013048528,
2.16328183754693498826531986391, 5.32230188850343063051485135572, 6.23647683668766188606777271392, 7.18550153265484235275048043276, 8.171367284506757795836711620592, 10.01388430107155303610249407887, 11.53069627997579327143350893621, 12.74797488785868980113236022850, 13.15108878847419480855728300746, 14.53861091016083444989859841512