L(s) = 1 | − i·2-s + 4-s − 3i·7-s − 3i·8-s + 2·11-s + 2i·13-s − 3·14-s − 16-s − 4i·17-s + 8·19-s − 2i·22-s + 3i·23-s + 2·26-s − 3i·28-s − 29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.5·4-s − 1.13i·7-s − 1.06i·8-s + 0.603·11-s + 0.554i·13-s − 0.801·14-s − 0.250·16-s − 0.970i·17-s + 1.83·19-s − 0.426i·22-s + 0.625i·23-s + 0.392·26-s − 0.566i·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.226823184\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.226823184\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.376741619708044686081009119557, −7.943172169087852326465739355537, −7.15597677106333918247873720315, −6.85542743206508358566982511708, −5.69687811853695285147992883523, −4.61755931869771490478403951750, −3.68037413688321536704585762883, −3.05726363888766807250643789256, −1.72838282775858378684817411189, −0.835991572129583043879403521465,
1.48514295959288926795682074441, 2.60315331284799429850266084139, 3.46407843211430183907768555112, 4.87293775156618201827285958311, 5.65579691115611211232008148623, 6.14943180023679515630897611102, 7.02473703030165987105606274057, 7.80973990263537273930878371406, 8.504759777416897739586786736708, 9.163013812254466075787784776182