L(s) = 1 | − i·2-s − 4-s − 1.84·5-s + i·8-s + 1.84i·10-s + 4i·11-s − 6.62i·13-s + 16-s − 2.29·17-s + 3.06i·19-s + 1.84·20-s + 4·22-s + 5.65i·23-s − 1.58·25-s − 6.62·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.826·5-s + 0.353i·8-s + 0.584i·10-s + 1.20i·11-s − 1.83i·13-s + 0.250·16-s − 0.556·17-s + 0.702i·19-s + 0.413·20-s + 0.852·22-s + 1.17i·23-s − 0.317·25-s − 1.29·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.630697 + 0.359838i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.630697 + 0.359838i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.84T + 5T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 6.62iT - 13T^{2} \) |
| 17 | \( 1 + 2.29T + 17T^{2} \) |
| 19 | \( 1 - 3.06iT - 19T^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 - 9.65iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 - 0.765T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 + 0.242iT - 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 7.97iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 2.34iT - 71T^{2} \) |
| 73 | \( 1 + 9.23iT - 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 3.11T + 89T^{2} \) |
| 97 | \( 1 - 9.23iT - 97T^{2} \) |
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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
−10.50437248544015324206424359807, −9.569135577217259224976355018938, −8.647695902478536389197970665712, −7.74820823500460695600565319034, −7.15937782838918978042203130833, −5.66643145452859518433514153309, −4.82790324702468427588706436561, −3.76439805507459780394058443818, −2.93269313950883052376490100919, −1.43278217748465579084951167867,
0.36659667843215848779998801125, 2.45606857955471988267594075019, 4.07790657329125424346364828910, 4.37303672961170537431765995603, 5.88917170240245915620287923757, 6.52330443787872329053463183579, 7.46888056351823500647371036193, 8.263627084270593752670531118098, 8.989819749795822863417346299580, 9.712235447545928403905914237390