L(s) = 1 | + (1.32 − 2.29i)2-s + (1.19 + 2.07i)3-s + (−2.52 − 4.37i)4-s + (1.82 − 3.16i)5-s + 6.36·6-s − 8.10·8-s + (−1.37 + 2.37i)9-s + (−4.85 − 8.40i)10-s + (−0.327 − 0.567i)11-s + (6.05 − 10.4i)12-s + 13-s + 8.75·15-s + (−5.70 + 9.88i)16-s + (1.19 + 2.07i)17-s + (3.63 + 6.30i)18-s + (−1.35 + 2.34i)19-s + ⋯ |
L(s) = 1 | + (0.938 − 1.62i)2-s + (0.691 + 1.19i)3-s + (−1.26 − 2.18i)4-s + (0.817 − 1.41i)5-s + 2.59·6-s − 2.86·8-s + (−0.456 + 0.791i)9-s + (−1.53 − 2.65i)10-s + (−0.0988 − 0.171i)11-s + (1.74 − 3.02i)12-s + 0.277·13-s + 2.26·15-s + (−1.42 + 2.47i)16-s + (0.290 + 0.503i)17-s + (0.857 + 1.48i)18-s + (−0.310 + 0.537i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33909 - 2.70126i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33909 - 2.70126i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-1.32 + 2.29i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.19 - 2.07i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.82 + 3.16i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.327 + 0.567i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.19 - 2.07i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.35 - 2.34i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.68 - 6.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.208T + 29T^{2} \) |
| 31 | \( 1 + (0.568 + 0.984i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.72 + 6.44i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 + (-2.30 + 3.98i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.62 + 4.55i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.12 - 7.15i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.948 - 1.64i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.75T + 71T^{2} \) |
| 73 | \( 1 + (-6.26 - 10.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.759 + 1.31i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + (7.40 - 12.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−10.16903479765052870592559821951, −9.669110103213298479157885117712, −9.092132218921289644163155241816, −8.271251076790563004188167919303, −5.78685725060117792074664621810, −5.37215047726054079495927231925, −4.20936416378128946540106961963, −3.83257658587863719919889198070, −2.47256243152806849617862955026, −1.31978081998146039087572749674,
2.39155427785449701940472030599, 3.18823143946371564697327092321, 4.62470290957021935952555385931, 6.01325808410659485710130239389, 6.45534730108446738766102693742, 7.15496490527734209652541508051, 7.79855568095427180711751615575, 8.628196276401745219391195245179, 9.731008586006313002753001951981, 10.98874369331260872979812931200