L(s) = 1 | + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.499 + 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)20-s + (−0.499 + 0.866i)25-s − 0.999·27-s + (−1 − 1.73i)29-s + 0.999·33-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.499 + 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)20-s + (−0.499 + 0.866i)25-s − 0.999·27-s + (−1 − 1.73i)29-s + 0.999·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.859352479\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.859352479\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.461649075383412604685670395989, −8.830634426764261253818250608788, −7.70542270478423696039592675332, −7.13084045293438558390788670561, −6.21716041625704478195306475335, −5.64593344389995164486883232500, −4.41722737769005268360835949646, −3.48998316761678450915348268612, −2.66113838759954693414542334677, −1.98867027134210920072959217847,
1.36147802593647645543938663821, 2.00982462259905650845547969586, 2.97826453295927527026953194208, 4.11143142151056214534921320114, 5.37423098090952954652379940149, 6.00636610709355256618885161749, 6.93219664597148579633567750759, 7.40442447092486514667129213305, 8.291246865939892314338234982898, 8.933404997016142708570739941536