| L(s) = 1 | − 1.22·5-s + (1.48 − 2.19i)7-s + 3.31i·11-s + 1.60i·13-s + 6.11·17-s + 1.35i·19-s + 1.11i·23-s − 3.50·25-s + 9.39i·29-s − 7.00i·31-s + (−1.81 + 2.67i)35-s − 11.7·37-s − 5.86·41-s + 8.58·43-s + 3.15·47-s + ⋯ |
| L(s) = 1 | − 0.546·5-s + (0.560 − 0.828i)7-s + 0.998i·11-s + 0.443i·13-s + 1.48·17-s + 0.310i·19-s + 0.232i·23-s − 0.701·25-s + 1.74i·29-s − 1.25i·31-s + (−0.306 + 0.452i)35-s − 1.92·37-s − 0.916·41-s + 1.30·43-s + 0.460·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.559402075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559402075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.48 + 2.19i)T \) |
| good | 5 | \( 1 + 1.22T + 5T^{2} \) |
| 11 | \( 1 - 3.31iT - 11T^{2} \) |
| 13 | \( 1 - 1.60iT - 13T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 19 | \( 1 - 1.35iT - 19T^{2} \) |
| 23 | \( 1 - 1.11iT - 23T^{2} \) |
| 29 | \( 1 - 9.39iT - 29T^{2} \) |
| 31 | \( 1 + 7.00iT - 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 5.86T + 41T^{2} \) |
| 43 | \( 1 - 8.58T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 - 0.539iT - 53T^{2} \) |
| 59 | \( 1 - 6.87T + 59T^{2} \) |
| 61 | \( 1 - 7.40iT - 61T^{2} \) |
| 67 | \( 1 - 5.74T + 67T^{2} \) |
| 71 | \( 1 - 4.81iT - 71T^{2} \) |
| 73 | \( 1 - 14.5iT - 73T^{2} \) |
| 79 | \( 1 + 1.15T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 4.04iT - 97T^{2} \) |
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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
−8.744941686956172872536002681889, −7.933348408666045631312581321546, −7.35042207655757133147906454221, −6.89996200203102270839603747976, −5.63470535191659646458260566783, −4.95999449419552300402979588919, −4.00918458552897082441577643013, −3.52650341241557917107225375233, −2.07823715155255153811558737721, −1.10735559059924976811372957357,
0.56332189148930149604175004552, 1.90793114043160071179117232255, 3.06756842061157404313324610261, 3.69594236367622402078240258277, 4.86268120029508249230441217409, 5.55627179163446285821588676427, 6.15705921826158496861514122473, 7.27115434314953922475997560281, 8.044108991502746456956313464689, 8.414289533307745428387365243230
