| L(s) = 1 | − 4.69·5-s − 3.50i·7-s − 15.3i·11-s + 1.31·13-s + 29.5·17-s − 4.35i·19-s − 39.1i·23-s − 3.00·25-s − 13.2·29-s + 18.9i·31-s + 16.4i·35-s + 14.7·37-s − 33.8·41-s + 36.4i·43-s − 39.0i·47-s + ⋯ |
| L(s) = 1 | − 0.938·5-s − 0.500i·7-s − 1.39i·11-s + 0.100·13-s + 1.73·17-s − 0.229i·19-s − 1.70i·23-s − 0.120·25-s − 0.456·29-s + 0.610i·31-s + 0.469i·35-s + 0.397·37-s − 0.825·41-s + 0.848i·43-s − 0.831i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.088611170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.088611170\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 + 4.69T + 25T^{2} \) |
| 7 | \( 1 + 3.50iT - 49T^{2} \) |
| 11 | \( 1 + 15.3iT - 121T^{2} \) |
| 13 | \( 1 - 1.31T + 169T^{2} \) |
| 17 | \( 1 - 29.5T + 289T^{2} \) |
| 23 | \( 1 + 39.1iT - 529T^{2} \) |
| 29 | \( 1 + 13.2T + 841T^{2} \) |
| 31 | \( 1 - 18.9iT - 961T^{2} \) |
| 37 | \( 1 - 14.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 36.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 39.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 69.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 32.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 69.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 88.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 93.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 67.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 79.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 41.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 94.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 134.T + 9.40e3T^{2} \) |
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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
−8.339030976800910953892761076430, −7.65461167569495899898207881172, −6.92750392575670105355242331260, −6.00180263993004966878122317867, −5.25544896253886174892889274152, −4.17230973950567255999208170280, −3.55464957019463561581543971219, −2.75980210825359234036823254406, −1.11883178722908233281967671872, −0.30180527423655070099597963398,
1.24460322356283425327750806793, 2.32207814585873692514153419569, 3.53407021153878755880338941081, 4.03556100283691272089410391584, 5.19975065724186630438499068786, 5.69153094072384391326259156823, 6.88804694267357693385340606107, 7.66425346595413934599199713575, 7.891175707952596927519611128909, 9.003332293897545403648688418674
