| L(s) = 1 | + (1.49 + 2.25i)2-s + (−2.07 + 4.90i)4-s + (3.86 + 1.23i)5-s + (−0.664 − 1.88i)7-s + (−8.85 + 1.65i)8-s + (3.00 + 10.5i)10-s + (−4.48 + 0.276i)11-s + (5.97 + 1.11i)13-s + (3.26 − 4.31i)14-s + (−9.55 − 9.85i)16-s + (−4.11 + 1.89i)17-s + (−4.30 + 2.31i)19-s + (−14.0 + 16.4i)20-s + (−7.32 − 9.70i)22-s + (1.49 + 2.99i)23-s + ⋯ |
| L(s) = 1 | + (1.05 + 1.59i)2-s + (−1.03 + 2.45i)4-s + (1.72 + 0.550i)5-s + (−0.251 − 0.712i)7-s + (−3.12 + 0.585i)8-s + (0.950 + 3.34i)10-s + (−1.35 + 0.0833i)11-s + (1.65 + 0.309i)13-s + (0.871 − 1.15i)14-s + (−2.38 − 2.46i)16-s + (−0.998 + 0.459i)17-s + (−0.987 + 0.530i)19-s + (−3.14 + 3.67i)20-s + (−1.56 − 2.06i)22-s + (0.311 + 0.624i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.104489 + 3.01160i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.104489 + 3.01160i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 + (-8.74 + 5.15i)T \) |
| good | 2 | \( 1 + (-1.49 - 2.25i)T + (-0.779 + 1.84i)T^{2} \) |
| 5 | \( 1 + (-3.86 - 1.23i)T + (4.08 + 2.88i)T^{2} \) |
| 7 | \( 1 + (0.664 + 1.88i)T + (-5.45 + 4.38i)T^{2} \) |
| 11 | \( 1 + (4.48 - 0.276i)T + (10.9 - 1.35i)T^{2} \) |
| 13 | \( 1 + (-5.97 - 1.11i)T + (12.1 + 4.69i)T^{2} \) |
| 17 | \( 1 + (4.11 - 1.89i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (4.30 - 2.31i)T + (10.4 - 15.8i)T^{2} \) |
| 23 | \( 1 + (-1.49 - 2.99i)T + (-13.8 + 18.3i)T^{2} \) |
| 29 | \( 1 + (0.0384 + 0.175i)T + (-26.3 + 12.1i)T^{2} \) |
| 31 | \( 1 + (-1.14 + 4.04i)T + (-26.3 - 16.3i)T^{2} \) |
| 37 | \( 1 + (-6.81 - 2.64i)T + (27.3 + 24.9i)T^{2} \) |
| 41 | \( 1 + (-6.60 + 2.09i)T + (33.4 - 23.6i)T^{2} \) |
| 43 | \( 1 + (-0.447 + 2.88i)T + (-40.9 - 13.0i)T^{2} \) |
| 47 | \( 1 + (-3.22 - 5.57i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.107 + 3.49i)T + (-52.8 - 3.26i)T^{2} \) |
| 59 | \( 1 + (-2.00 + 5.68i)T + (-45.9 - 36.9i)T^{2} \) |
| 61 | \( 1 + (-1.25 + 13.5i)T + (-59.9 - 11.2i)T^{2} \) |
| 67 | \( 1 + (-1.54 - 1.79i)T + (-10.2 + 66.2i)T^{2} \) |
| 71 | \( 1 + (0.592 - 2.70i)T + (-64.5 - 29.6i)T^{2} \) |
| 73 | \( 1 + (5.54 + 5.05i)T + (6.73 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-3.21 + 2.93i)T + (7.28 - 78.6i)T^{2} \) |
| 83 | \( 1 + (-6.44 + 7.52i)T + (-12.7 - 82.0i)T^{2} \) |
| 89 | \( 1 + (-0.235 + 0.312i)T + (-24.3 - 85.6i)T^{2} \) |
| 97 | \( 1 + (4.12 + 1.89i)T + (63.1 + 73.6i)T^{2} \) |
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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
−10.48011705707016541268199313150, −9.416582372920357333917623813443, −8.548404702948681501041987164769, −7.68006227149471763847425541413, −6.66682706818631207224629401725, −6.16997971982603922114816306753, −5.63471677575530077449432609168, −4.52344122135517596610307603146, −3.53324545162426055947716180207, −2.29856113028700800093492255788,
1.04930971615361034876586246370, 2.42532014629804068998614649545, 2.65358271278058327991074049422, 4.28697509866628545796921330931, 5.16655753008281567751793965192, 5.84278343506712962661203996012, 6.37024311854936976918216010995, 8.725393762297618265703438408378, 8.991090704617202505645614251354, 9.981129883156055050640290474623
