L(s) = 1 | + (1.35 + 0.402i)2-s + (−0.631 + 0.631i)3-s + (1.67 + 1.09i)4-s + (2.34 + 2.34i)5-s + (−1.10 + 0.601i)6-s + (1.83 + 2.15i)8-s + 2.20i·9-s + (2.23 + 4.11i)10-s + (−2.18 − 2.18i)11-s + (−1.74 + 0.368i)12-s + (4.03 − 4.03i)13-s − 2.95·15-s + (1.61 + 3.65i)16-s − 0.347·17-s + (−0.886 + 2.98i)18-s + (−4.26 + 4.26i)19-s + ⋯ |
L(s) = 1 | + (0.958 + 0.284i)2-s + (−0.364 + 0.364i)3-s + (0.837 + 0.545i)4-s + (1.04 + 1.04i)5-s + (−0.453 + 0.245i)6-s + (0.647 + 0.761i)8-s + 0.734i·9-s + (0.706 + 1.30i)10-s + (−0.658 − 0.658i)11-s + (−0.504 + 0.106i)12-s + (1.11 − 1.11i)13-s − 0.763·15-s + (0.404 + 0.914i)16-s − 0.0843·17-s + (−0.209 + 0.704i)18-s + (−0.978 + 0.978i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0234 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0234 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.03740 + 2.08579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03740 + 2.08579i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.35 - 0.402i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.631 - 0.631i)T - 3iT^{2} \) |
| 5 | \( 1 + (-2.34 - 2.34i)T + 5iT^{2} \) |
| 11 | \( 1 + (2.18 + 2.18i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.03 + 4.03i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.347T + 17T^{2} \) |
| 19 | \( 1 + (4.26 - 4.26i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.23iT - 23T^{2} \) |
| 29 | \( 1 + (-1.21 + 1.21i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + (6.42 + 6.42i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.68iT - 41T^{2} \) |
| 43 | \( 1 + (-4.05 - 4.05i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.64T + 47T^{2} \) |
| 53 | \( 1 + (-8.44 - 8.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.17 - 5.17i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.00533 + 0.00533i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3.02 + 3.02i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.828iT - 71T^{2} \) |
| 73 | \( 1 + 6.25iT - 73T^{2} \) |
| 79 | \( 1 + 0.755T + 79T^{2} \) |
| 83 | \( 1 + (-3.66 + 3.66i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.24iT - 89T^{2} \) |
| 97 | \( 1 + 2.18T + 97T^{2} \) |
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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
−10.67727471388445205180575181610, −10.25286019905630705637163812552, −8.552602677526238912251163608042, −7.84304205150988439784718113786, −6.64539827918290919196057153754, −5.87247673566179829607648477401, −5.49956321292900100855978132293, −4.18190917082137366369724428253, −3.02835655042583883039801869893, −2.16094457043165057572197527955,
1.25346595960211204817691241437, 2.14676219387995226788234611697, 3.73673681009830920783211895451, 4.80076384645953109121216428942, 5.51748659836550032531049279132, 6.42638369464586253658678164184, 6.98684059878427044674195995117, 8.543588533880459901587720345479, 9.363400299746065035566307715616, 10.11350885603332964093709487380