L(s) = 1 | + (−1.13 − 2.77i)3-s + 5.85i·7-s + (−6.42 + 6.29i)9-s + 12.4i·11-s − 11.8i·13-s − 29.2·17-s + 3.19·19-s + (16.2 − 6.64i)21-s + 19.5·23-s + (24.7 + 10.7i)27-s − 30.3i·29-s − 3.57·31-s + (34.6 − 14.1i)33-s − 42.7i·37-s + (−32.9 + 13.4i)39-s + ⋯ |
L(s) = 1 | + (−0.377 − 0.925i)3-s + 0.836i·7-s + (−0.714 + 0.699i)9-s + 1.13i·11-s − 0.912i·13-s − 1.72·17-s + 0.168·19-s + (0.774 − 0.316i)21-s + 0.849·23-s + (0.917 + 0.396i)27-s − 1.04i·29-s − 0.115·31-s + (1.04 − 0.428i)33-s − 1.15i·37-s + (−0.844 + 0.344i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0760 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0760 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.139871066\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.139871066\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.13 + 2.77i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 5.85iT - 49T^{2} \) |
| 11 | \( 1 - 12.4iT - 121T^{2} \) |
| 13 | \( 1 + 11.8iT - 169T^{2} \) |
| 17 | \( 1 + 29.2T + 289T^{2} \) |
| 19 | \( 1 - 3.19T + 361T^{2} \) |
| 23 | \( 1 - 19.5T + 529T^{2} \) |
| 29 | \( 1 + 30.3iT - 841T^{2} \) |
| 31 | \( 1 + 3.57T + 961T^{2} \) |
| 37 | \( 1 + 42.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 6.39iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 62.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 69.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 57.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 78.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 68.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 90.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 26.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 40.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 148.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 9.36T + 6.88e3T^{2} \) |
| 89 | \( 1 + 109. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 161. iT - 9.40e3T^{2} \) |
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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
−9.091195254171283384185375006497, −8.603560355424695520136288451594, −7.46026508757393507182010821853, −7.00593909729700961231962321361, −5.94324193524043637410285234388, −5.30705055726995742362767014385, −4.24893541414998419895063403882, −2.61955850508013348475784256792, −2.02319626438183584861793511163, −0.43068518991225412591775949330,
0.970188433539684069330006500584, 2.75754283063273541295135437001, 3.84467761128851770982405048223, 4.49962265133929892508810111819, 5.43030406224390516277540367399, 6.49499036269605690524519991468, 7.04691403954840336908617628893, 8.481308463111359930784201406128, 8.941322170740559647418580796812, 9.805559140700682644531725737024