L(s) = 1 | − 6i·5-s − 5i·7-s + 6·11-s + 5.19i·13-s + 31.1·17-s + 25.9·19-s + 31.1i·23-s − 11·25-s + 36i·29-s + 26i·31-s − 30·35-s − 25.9i·37-s + 20.7·43-s + 31.1i·47-s + 24·49-s + ⋯ |
L(s) = 1 | − 1.20i·5-s − 0.714i·7-s + 0.545·11-s + 0.399i·13-s + 1.83·17-s + 1.36·19-s + 1.35i·23-s − 0.440·25-s + 1.24i·29-s + 0.838i·31-s − 0.857·35-s − 0.702i·37-s + 0.483·43-s + 0.663i·47-s + 0.489·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.417313529\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.417313529\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 6iT - 25T^{2} \) |
| 7 | \( 1 + 5iT - 49T^{2} \) |
| 11 | \( 1 - 6T + 121T^{2} \) |
| 13 | \( 1 - 5.19iT - 169T^{2} \) |
| 17 | \( 1 - 31.1T + 289T^{2} \) |
| 19 | \( 1 - 25.9T + 361T^{2} \) |
| 23 | \( 1 - 31.1iT - 529T^{2} \) |
| 29 | \( 1 - 36iT - 841T^{2} \) |
| 31 | \( 1 - 26iT - 961T^{2} \) |
| 37 | \( 1 + 25.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 60iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 18T + 3.48e3T^{2} \) |
| 61 | \( 1 + 77.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 77.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 25T + 5.32e3T^{2} \) |
| 79 | \( 1 + 31iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 120T + 6.88e3T^{2} \) |
| 89 | \( 1 + 155.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 85T + 9.40e3T^{2} \) |
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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
−9.289530934576666150586455257090, −8.182566431503171580453455295197, −7.54114917531752628299064801723, −6.77800733108311039968678163274, −5.43023547581938228249597412206, −5.16808832369739981574809079486, −3.93017100228013014441800575752, −3.29491514078684583148219796061, −1.45828186040346089790104176557, −0.940128755938611742621601949487,
0.948337426413785936906888480537, 2.48681792633956190314374389686, 3.09858408023373421282963480037, 4.07572387768401907297457261914, 5.41025967108016951585593370322, 5.97150597631172319175500137879, 6.85587004359099021452203561961, 7.63928246961121603554995764378, 8.316850758658861840078590639404, 9.414791706230470417492518969654