L(s) = 1 | + (1.20 + 2.09i)2-s − 1.41·3-s + (−1.91 + 3.31i)4-s + (−1.91 + 3.31i)5-s + (−1.70 − 2.95i)6-s − 4.41·8-s − 0.999·9-s − 9.24·10-s + 3.41·11-s + (2.70 − 4.68i)12-s + (−3.5 − 0.866i)13-s + (2.70 − 4.68i)15-s + (−1.49 − 2.59i)16-s + (0.0857 − 0.148i)17-s + (−1.20 − 2.09i)18-s + 6·19-s + ⋯ |
L(s) = 1 | + (0.853 + 1.47i)2-s − 0.816·3-s + (−0.957 + 1.65i)4-s + (−0.856 + 1.48i)5-s + (−0.696 − 1.20i)6-s − 1.56·8-s − 0.333·9-s − 2.92·10-s + 1.02·11-s + (0.781 − 1.35i)12-s + (−0.970 − 0.240i)13-s + (0.698 − 1.21i)15-s + (−0.374 − 0.649i)16-s + (0.0208 − 0.0360i)17-s + (−0.284 − 0.492i)18-s + 1.37·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.593786 - 0.665603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.593786 - 0.665603i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1.20 - 2.09i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + (1.91 - 3.31i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 17 | \( 1 + (-0.0857 + 0.148i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + (0.707 + 1.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.91 - 8.51i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.70 + 4.68i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.74 + 6.48i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.91 - 5.04i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.292 + 0.507i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.82 - 6.63i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.878 + 1.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + (-0.171 - 0.297i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.328 + 0.568i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.12 - 8.87i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + (-3.65 - 6.33i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.58 - 4.47i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.44287059883156436550360901508, −10.58933809490194377186453139362, −9.327481877734618195698970770203, −8.052253923369356853386892702374, −7.19505016904438601886433281364, −6.86209090588049083146662626683, −5.89272355207702351268705846897, −5.11977815984498563838506659718, −3.92971702860887586271467173199, −3.08144851496410762630810795252,
0.41905424541474725897567591108, 1.62759586374102032425066402327, 3.35898316871274699661793414253, 4.28260351428739860819926292435, 5.06041902576579618316353196856, 5.65163846336988181151201707159, 7.20816186267345972466884967230, 8.500280742828886589662518294693, 9.396326286772014309728901758357, 10.12019836406894005073649067390