L(s) = 1 | + (1.39 − 0.245i)3-s + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)7-s + (0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (0.909 − 1.08i)15-s + (−0.939 − 0.342i)17-s + (−0.483 + 1.32i)21-s + (−0.483 − 1.32i)29-s + (0.909 + 1.08i)33-s + (0.173 + 0.984i)35-s − 1.41i·37-s + (−1.39 + 0.245i)41-s + (−0.766 + 0.642i)43-s + (0.5 − 0.866i)45-s + ⋯ |
L(s) = 1 | + (1.39 − 0.245i)3-s + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)7-s + (0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (0.909 − 1.08i)15-s + (−0.939 − 0.342i)17-s + (−0.483 + 1.32i)21-s + (−0.483 − 1.32i)29-s + (0.909 + 1.08i)33-s + (0.173 + 0.984i)35-s − 1.41i·37-s + (−1.39 + 0.245i)41-s + (−0.766 + 0.642i)43-s + (0.5 − 0.866i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.807746979\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.807746979\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-1.39 + 0.245i)T + (0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.483 + 1.32i)T + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 + (1.39 - 0.245i)T + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.483 + 1.32i)T + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (0.909 + 1.08i)T + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
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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
−9.350883675468038078267804840865, −9.096602846065125961313591330332, −8.315453583665843828004193965032, −7.40168942618768218280571012343, −6.51638618379285826799834970332, −5.60015685553848543381938974741, −4.57302430279781474371595263329, −3.50086115837026614843539293411, −2.35591308637796925960359329067, −1.85495336278615597046951111113,
1.70804044143355649779701917753, 2.87335052574645884269982823555, 3.48396566132526397450307567168, 4.35381267001590532461947186799, 5.76926113466749525243536006540, 6.69391169971898602250802923910, 7.22966714847943927310351654226, 8.469830036349598499594503391280, 8.813901710550359551431783122463, 9.743505173051954444042567975374