L(s) = 1 | + (0.965 − 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s − i·6-s + (−0.965 − 2.46i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.55 − 2.69i)11-s + (−0.258 − 0.965i)12-s + (−2.30 − 2.30i)13-s + (−1.57 − 2.12i)14-s + (0.500 − 0.866i)16-s + (−1.10 − 0.295i)17-s + (−0.965 − 0.258i)18-s + (−2.99 + 5.18i)19-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s − 0.408i·6-s + (−0.365 − 0.930i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.469 − 0.813i)11-s + (−0.0747 − 0.278i)12-s + (−0.639 − 0.639i)13-s + (−0.419 − 0.569i)14-s + (0.125 − 0.216i)16-s + (−0.267 − 0.0716i)17-s + (−0.227 − 0.0610i)18-s + (−0.687 + 1.19i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.877647433\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.877647433\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.965 + 2.46i)T \) |
good | 11 | \( 1 + (1.55 + 2.69i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.30 + 2.30i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.10 + 0.295i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (2.99 - 5.18i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.295 + 1.10i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 5.39iT - 29T^{2} \) |
| 31 | \( 1 + (-3.68 + 2.12i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.55 + 1.48i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (-6.44 + 6.44i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.10 + 4.11i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.78 + 0.746i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.117 - 0.203i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.38 - 4.26i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.60 + 13.4i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 1.74T + 71T^{2} \) |
| 73 | \( 1 + (0.293 - 1.09i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (8.76 + 5.05i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.06 - 8.06i)T + 83iT^{2} \) |
| 89 | \( 1 + (9.18 - 15.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.740 - 0.740i)T - 97iT^{2} \) |
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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
−9.829086182708483003046639469443, −8.592956074093636627115453595063, −7.77461654618804359034959142910, −7.03434980523291563532089158997, −6.13909240094417215860667755106, −5.34202241185602537203482930390, −4.12824599240650123916855381662, −3.27776649915854944399933418559, −2.20099463937572447255767113825, −0.62132031065853336761966040006,
2.26994971355922840723291174498, 2.90268154293584660817482650919, 4.41409189443377479578771801340, 4.77590100245057820441005385922, 5.93942759856759377839508076204, 6.67866770576280807197271521137, 7.69256425147076277275594478501, 8.624418525800241692732603940048, 9.519325160837871816039376035486, 10.04694517266541688112832704733