L(s) = 1 | + (−1.10 + 0.880i)2-s + (0.450 − 1.94i)4-s + (2.03 − 1.17i)5-s + (−2.03 − 1.69i)7-s + (1.21 + 2.55i)8-s + (−1.21 + 3.08i)10-s + (2.18 + 1.26i)11-s + 1.48i·13-s + (3.74 + 0.0828i)14-s + (−3.59 − 1.75i)16-s + (1.66 + 0.958i)17-s + (0.454 + 0.786i)19-s + (−1.36 − 4.48i)20-s + (−3.53 + 0.526i)22-s + (4.55 − 2.63i)23-s + ⋯ |
L(s) = 1 | + (−0.782 + 0.622i)2-s + (0.225 − 0.974i)4-s + (0.908 − 0.524i)5-s + (−0.768 − 0.639i)7-s + (0.429 + 0.902i)8-s + (−0.384 + 0.976i)10-s + (0.659 + 0.380i)11-s + 0.411i·13-s + (0.999 + 0.0221i)14-s + (−0.898 − 0.439i)16-s + (0.402 + 0.232i)17-s + (0.104 + 0.180i)19-s + (−0.306 − 1.00i)20-s + (−0.753 + 0.112i)22-s + (0.950 − 0.548i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15633 - 0.157583i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15633 - 0.157583i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.10 - 0.880i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.03 + 1.69i)T \) |
good | 5 | \( 1 + (-2.03 + 1.17i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.18 - 1.26i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 1.48iT - 13T^{2} \) |
| 17 | \( 1 + (-1.66 - 0.958i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.454 - 0.786i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.55 + 2.63i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + (-3.77 + 6.53i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.63 + 8.03i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 12.0iT - 41T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 + (-2.04 - 3.54i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.83 - 4.91i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.98 - 6.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.72 + 3.30i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.36 - 0.790i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.670iT - 71T^{2} \) |
| 73 | \( 1 + (-2.49 - 1.44i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.5 + 7.81i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 + (-3.23 + 1.86i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14.0iT - 97T^{2} \) |
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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
−10.17355781511128562016408162487, −9.209829454266864740960501495175, −8.969676504533815681624788272142, −7.61996775958729532618380241026, −6.83585908782705148828478208420, −6.09727271007573833464621986875, −5.18959065576619625157598296326, −3.97640860950065430572251881832, −2.19775156929551303479410552459, −0.899773778282652751113611434514,
1.30295161285090989301125406482, 2.78795216360779443459959135879, 3.27299650230962558914159535188, 4.98755744613734786981934572210, 6.34678937953875630053384052831, 6.72352115146972987169512999478, 8.079908025124231143734719868830, 8.876663815215062921409830702774, 9.772284530572939887976628538637, 10.05096366189085424773059730402