L(s) = 1 | + (−0.0941 + 1.41i)2-s + (−0.416 + 1.68i)3-s + (−1.98 − 0.265i)4-s + (1.62 + 1.54i)5-s + (−2.33 − 0.746i)6-s + (−0.361 − 0.361i)7-s + (0.561 − 2.77i)8-s + (−2.65 − 1.40i)9-s + (−2.32 + 2.14i)10-s − 2.63·11-s + (1.27 − 3.22i)12-s + (3.49 + 3.49i)13-s + (0.544 − 0.476i)14-s + (−3.26 + 2.08i)15-s + (3.85 + 1.05i)16-s + (3.61 − 3.61i)17-s + ⋯ |
L(s) = 1 | + (−0.0665 + 0.997i)2-s + (−0.240 + 0.970i)3-s + (−0.991 − 0.132i)4-s + (0.724 + 0.688i)5-s + (−0.952 − 0.304i)6-s + (−0.136 − 0.136i)7-s + (0.198 − 0.980i)8-s + (−0.884 − 0.466i)9-s + (−0.735 + 0.677i)10-s − 0.794·11-s + (0.367 − 0.930i)12-s + (0.968 + 0.968i)13-s + (0.145 − 0.127i)14-s + (−0.842 + 0.538i)15-s + (0.964 + 0.263i)16-s + (0.876 − 0.876i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.294231 + 0.874608i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.294231 + 0.874608i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0941 - 1.41i)T \) |
| 3 | \( 1 + (0.416 - 1.68i)T \) |
| 5 | \( 1 + (-1.62 - 1.54i)T \) |
good | 7 | \( 1 + (0.361 + 0.361i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 + (-3.49 - 3.49i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.61 + 3.61i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.672T + 19T^{2} \) |
| 23 | \( 1 + (-4.31 - 4.31i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.76iT - 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 + (2.82 - 2.82i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.10iT - 41T^{2} \) |
| 43 | \( 1 + (7.57 + 7.57i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.987 + 0.987i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.646 - 0.646i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.92iT - 59T^{2} \) |
| 61 | \( 1 + 6.07iT - 61T^{2} \) |
| 67 | \( 1 + (0.349 - 0.349i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.63iT - 71T^{2} \) |
| 73 | \( 1 + (11.3 - 11.3i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.07iT - 79T^{2} \) |
| 83 | \( 1 + (-8.53 + 8.53i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.58T + 89T^{2} \) |
| 97 | \( 1 + (0.660 + 0.660i)T + 97iT^{2} \) |
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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
−13.96209056768667841840436490781, −13.45383738973057389083129602606, −11.63524942833902705227138812523, −10.41063916130411454167731520132, −9.685707952967844404653724258956, −8.679519008667815073186260562454, −7.14715141469026893008876660705, −6.01259595739460948102861256608, −5.03717212265132506866764312114, −3.45276236132942520383021396879,
1.25486116545243373921808275420, 2.91984569372282031261810781133, 5.08717132526540000131402852927, 6.07525383920787500569993230532, 8.032375744686342222596686895644, 8.724018154922857901939226054798, 10.18746332368654384452540796957, 10.98426605801230822035672887092, 12.39946957281253113651886712658, 12.87337239476214164909556065430