| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 3.37i·7-s − i·8-s − 9-s − 1.37·11-s + i·12-s − 2i·13-s − 3.37·14-s + 16-s − i·17-s − i·18-s − 3.37·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.27i·7-s − 0.353i·8-s − 0.333·9-s − 0.413·11-s + 0.288i·12-s − 0.554i·13-s − 0.901·14-s + 0.250·16-s − 0.242i·17-s − 0.235i·18-s − 0.773·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9086280021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9086280021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 - 3.37iT - 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 + 4.37iT - 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 + 8.11T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 + 9.37iT - 43T^{2} \) |
| 47 | \( 1 + 7.37iT - 47T^{2} \) |
| 53 | \( 1 - 5.74iT - 53T^{2} \) |
| 59 | \( 1 - 7.11T + 59T^{2} \) |
| 61 | \( 1 - 9.11T + 61T^{2} \) |
| 67 | \( 1 + 5.37iT - 67T^{2} \) |
| 71 | \( 1 - 4.37T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 6.11T + 79T^{2} \) |
| 83 | \( 1 - 4.37iT - 83T^{2} \) |
| 89 | \( 1 + 8.74T + 89T^{2} \) |
| 97 | \( 1 + 1.25iT - 97T^{2} \) |
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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
−8.642695658898264838396242857778, −8.053808910276292519097148777344, −7.18346776044950068920349282853, −6.48988476649999932679678624958, −5.61248554198752612652071989732, −5.25888771762156118942773963127, −4.04914355762185485333017565869, −2.82586232763729123950551301710, −2.03420409575606134297047951960, −0.32445121078107987619924142662,
1.15170760478823635423787717326, 2.36711450068850214459224623059, 3.52450324864135045525625314546, 4.09731127560463502513090494163, 4.81637470674570990891967353079, 5.77068879690959970073818222810, 6.80023349395969865566267926363, 7.61171933261582685030063770704, 8.391017721208264395389550093674, 9.244750983011073717381501716750
