| L(s) = 1 | − 3.20i·3-s + 3.06·5-s + 12.3·7-s − 1.30·9-s − 5.53·11-s − 6.70i·13-s − 9.83i·15-s − 5.36·17-s + (1.17 − 18.9i)19-s − 39.5i·21-s + 9.72·23-s − 15.6·25-s − 24.7i·27-s + 14.4i·29-s + 36.0i·31-s + ⋯ |
| L(s) = 1 | − 1.06i·3-s + 0.612·5-s + 1.75·7-s − 0.144·9-s − 0.503·11-s − 0.516i·13-s − 0.655i·15-s − 0.315·17-s + (0.0618 − 0.998i)19-s − 1.88i·21-s + 0.422·23-s − 0.624·25-s − 0.915i·27-s + 0.497i·29-s + 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.712009089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.712009089\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.17 + 18.9i)T \) |
| good | 3 | \( 1 + 3.20iT - 9T^{2} \) |
| 5 | \( 1 - 3.06T + 25T^{2} \) |
| 7 | \( 1 - 12.3T + 49T^{2} \) |
| 11 | \( 1 + 5.53T + 121T^{2} \) |
| 13 | \( 1 + 6.70iT - 169T^{2} \) |
| 17 | \( 1 + 5.36T + 289T^{2} \) |
| 23 | \( 1 - 9.72T + 529T^{2} \) |
| 29 | \( 1 - 14.4iT - 841T^{2} \) |
| 31 | \( 1 - 36.0iT - 961T^{2} \) |
| 37 | \( 1 + 32.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 60.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 58.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 24.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 107. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 45.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 42.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 70.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 52.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 2.68iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 138.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 106. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 79.3iT - 9.40e3T^{2} \) |
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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
−9.044678325673354390255552562949, −8.471123246032075036286337515210, −7.41172742709992774098248320816, −7.24608652994631883238062560148, −5.88381901967067845354322933862, −5.22436525048594138526353405833, −4.29352498243230217113396857574, −2.58359442649451443574026750871, −1.83261900932631507685188081052, −0.853273258687120880740524562002,
1.42330530743012649154596567791, 2.40538837985337403047311748312, 3.94836442478158605962527275236, 4.59937704861255661496544111516, 5.31767577232109144273096411554, 6.17538172885581138703316887292, 7.56563380237947923401071436407, 8.113952115484033476035659190010, 9.121074094310069119178986373516, 9.765902301124595773042748229615
