L(s) = 1 | + (0.546 + 0.546i)2-s − 0.487i·3-s − 1.40i·4-s + (1.29 − 1.29i)5-s + (0.266 − 0.266i)6-s + (1.85 − 1.85i)8-s + 2.76·9-s + 1.41·10-s + (−0.725 + 0.725i)11-s − 0.683·12-s + (−0.266 − 3.59i)13-s + (−0.628 − 0.628i)15-s − 0.773·16-s − 5.20·17-s + (1.50 + 1.50i)18-s + (−3.71 + 3.71i)19-s + ⋯ |
L(s) = 1 | + (0.386 + 0.386i)2-s − 0.281i·3-s − 0.701i·4-s + (0.577 − 0.577i)5-s + (0.108 − 0.108i)6-s + (0.657 − 0.657i)8-s + 0.920·9-s + 0.445·10-s + (−0.218 + 0.218i)11-s − 0.197·12-s + (−0.0738 − 0.997i)13-s + (−0.162 − 0.162i)15-s − 0.193·16-s − 1.26·17-s + (0.355 + 0.355i)18-s + (−0.852 + 0.852i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75031 - 1.04582i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75031 - 1.04582i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (0.266 + 3.59i)T \) |
good | 2 | \( 1 + (-0.546 - 0.546i)T + 2iT^{2} \) |
| 3 | \( 1 + 0.487iT - 3T^{2} \) |
| 5 | \( 1 + (-1.29 + 1.29i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.725 - 0.725i)T - 11iT^{2} \) |
| 17 | \( 1 + 5.20T + 17T^{2} \) |
| 19 | \( 1 + (3.71 - 3.71i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.843iT - 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + (-4.16 + 4.16i)T - 31iT^{2} \) |
| 37 | \( 1 + (-4.41 + 4.41i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.0927 - 0.0927i)T - 41iT^{2} \) |
| 43 | \( 1 - 7.36iT - 43T^{2} \) |
| 47 | \( 1 + (1.59 + 1.59i)T + 47iT^{2} \) |
| 53 | \( 1 + 6.77T + 53T^{2} \) |
| 59 | \( 1 + (-7.12 - 7.12i)T + 59iT^{2} \) |
| 61 | \( 1 + 1.30iT - 61T^{2} \) |
| 67 | \( 1 + (-3.04 - 3.04i)T + 67iT^{2} \) |
| 71 | \( 1 + (6.02 + 6.02i)T + 71iT^{2} \) |
| 73 | \( 1 + (-8.02 - 8.02i)T + 73iT^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + (4.16 - 4.16i)T - 83iT^{2} \) |
| 89 | \( 1 + (-5.48 - 5.48i)T + 89iT^{2} \) |
| 97 | \( 1 + (-2.49 + 2.49i)T - 97iT^{2} \) |
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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
−10.19486620281165856679156132796, −9.820416634265258778655388322329, −8.650731785170831853362108544669, −7.67796525284293446618570288300, −6.59859959109181701999531694922, −5.99768789736344247872786201774, −4.90672916439851463697088340978, −4.25081006623047643108265070592, −2.29794541273832608792146566489, −1.06445673402800535356222200329,
2.02828627254681425335286164149, 2.93526063871829059700012341682, 4.30913401159770510682097133274, 4.74687487832657431318160321589, 6.55101472154905794327488016224, 6.87460042974464270057236188575, 8.231603867519221887978993619368, 9.004660479127969225547715792991, 10.08097302455907416601606306488, 10.75811779778568712193954492185