| L(s) = 1 | + i·3-s + (2.05 − 0.874i)5-s − 1.46i·7-s − 9-s − 11-s + 5.53i·13-s + (0.874 + 2.05i)15-s + 8.15i·17-s − 2.94·19-s + 1.46·21-s − 2.87i·23-s + (3.47 − 3.59i)25-s − i·27-s − 5.70·29-s − 5.49·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.920 − 0.391i)5-s − 0.552i·7-s − 0.333·9-s − 0.301·11-s + 1.53i·13-s + (0.225 + 0.531i)15-s + 1.97i·17-s − 0.674·19-s + 0.319·21-s − 0.600i·23-s + (0.694 − 0.719i)25-s − 0.192i·27-s − 1.05·29-s − 0.987·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.486332634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486332634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2.05 + 0.874i)T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 1.46iT - 7T^{2} \) |
| 13 | \( 1 - 5.53iT - 13T^{2} \) |
| 17 | \( 1 - 8.15iT - 17T^{2} \) |
| 19 | \( 1 + 2.94T + 19T^{2} \) |
| 23 | \( 1 + 2.87iT - 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 5.49T + 31T^{2} \) |
| 37 | \( 1 - 7.86iT - 37T^{2} \) |
| 41 | \( 1 + 0.761T + 41T^{2} \) |
| 43 | \( 1 - 0.841iT - 43T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 + 8.00iT - 53T^{2} \) |
| 59 | \( 1 - 6.57T + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 - 1.44iT - 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.07iT - 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 6.51iT - 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 0.207iT - 97T^{2} \) |
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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
−9.081147459942494909303490448272, −8.588153124729839814509972908225, −7.66908882832736871977178637702, −6.50809950357748206193295869605, −6.15567502335441147352530107679, −5.11353536121034264116555574582, −4.31944811116694726235833539026, −3.70899097570956985620807427857, −2.27264046291933202570993152151, −1.48841551238601574533972574229,
0.45480941520806087025396194952, 1.96308132816294237342821310936, 2.65779321781705540303314056779, 3.48248131124014224548678469650, 5.11913338556362745640295749315, 5.51895905199646307134677766594, 6.19993483679398107505881519510, 7.40262589654332195007549130443, 7.46753597607230656449369466462, 8.819568448837485869362884025376
