L(s) = 1 | + 2-s + (0.420 + 1.68i)3-s + 4-s + 3.36i·5-s + (0.420 + 1.68i)6-s + (2.37 − 1.16i)7-s + 8-s + (−2.64 + 1.41i)9-s + 3.36i·10-s + (0.420 + 1.68i)12-s + (−2.79 − 2.27i)13-s + (2.37 − 1.16i)14-s + (−5.64 + 1.41i)15-s + 16-s + 7.82·17-s + (−2.64 + 1.41i)18-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.242 + 0.970i)3-s + 0.5·4-s + 1.50i·5-s + (0.171 + 0.685i)6-s + (0.898 − 0.439i)7-s + 0.353·8-s + (−0.881 + 0.471i)9-s + 1.06i·10-s + (0.121 + 0.485i)12-s + (−0.775 − 0.631i)13-s + (0.635 − 0.311i)14-s + (−1.45 + 0.365i)15-s + 0.250·16-s + 1.89·17-s + (−0.623 + 0.333i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0179 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0179 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.72468 + 1.75591i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72468 + 1.75591i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.420 - 1.68i)T \) |
| 7 | \( 1 + (-2.37 + 1.16i)T \) |
| 13 | \( 1 + (2.79 + 2.27i)T \) |
good | 5 | \( 1 - 3.36iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 7.82T + 17T^{2} \) |
| 19 | \( 1 + 5.59T + 19T^{2} \) |
| 23 | \( 1 + 0.500iT - 23T^{2} \) |
| 29 | \( 1 + 5.15iT - 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + 2.32iT - 37T^{2} \) |
| 41 | \( 1 - 9.87iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 4.33iT - 47T^{2} \) |
| 53 | \( 1 + 0.500iT - 53T^{2} \) |
| 59 | \( 1 + 2.16iT - 59T^{2} \) |
| 61 | \( 1 - 4.55iT - 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 0.708T + 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 3.14iT - 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−10.86594669906221360717023471263, −10.41022646194682906230407080332, −9.646033464796044553947535224953, −8.010689368999383434167241600007, −7.57031332206715839726420363776, −6.27111190366325200432948894749, −5.32630506671858817082303792858, −4.26993994497284867154739850293, −3.32161872110317103064426462375, −2.39596921077032745336579671841,
1.26678522265415791953678982884, 2.29242286878425419671947571755, 3.95464164930227558061773156770, 5.15554748690114524894287604568, 5.62957038901126388328901625048, 7.00229672427072776411756769395, 7.933946874840838441613128566635, 8.589843072029457245480849805136, 9.445352933014817954611494375851, 10.88622034364657183804022532616