L(s) = 1 | + 3.12i·3-s + (1.32 − 1.80i)5-s + i·7-s − 6.76·9-s − 2.48·11-s + 4.15i·13-s + (5.64 + 4.12i)15-s + 5.76i·17-s − 1.60·19-s − 3.12·21-s + 7.28i·23-s + (−1.51 − 4.76i)25-s − 11.7i·27-s − 1.45·29-s + 2.24·31-s + ⋯ |
L(s) = 1 | + 1.80i·3-s + (0.590 − 0.807i)5-s + 0.377i·7-s − 2.25·9-s − 0.749·11-s + 1.15i·13-s + (1.45 + 1.06i)15-s + 1.39i·17-s − 0.369·19-s − 0.681·21-s + 1.51i·23-s + (−0.303 − 0.952i)25-s − 2.26i·27-s − 0.270·29-s + 0.404·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.391999 + 1.20002i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.391999 + 1.20002i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.32 + 1.80i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 3.12iT - 3T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 - 4.15iT - 13T^{2} \) |
| 17 | \( 1 - 5.76iT - 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 - 7.28iT - 23T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 5.28iT - 43T^{2} \) |
| 47 | \( 1 + 3.45iT - 47T^{2} \) |
| 53 | \( 1 - 9.21iT - 53T^{2} \) |
| 59 | \( 1 + 5.92T + 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 67 | \( 1 - 7.52iT - 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 7.28iT - 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 2.73iT - 97T^{2} \) |
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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
−10.83580734695944948155423385135, −10.19665192361274187024739870039, −9.230164618047807929472459195072, −8.991600323536996309795660354084, −7.891246890773051475159617723420, −6.06972711139620324632373984255, −5.44657680420239610876292575015, −4.50790816033526812573190738198, −3.72240456151987258086946270619, −2.16402511441150797442928135567,
0.70244799795944258119014251231, 2.35594546444084911528761148237, 2.96487915006744858357021861503, 5.10682264309828942400842904404, 6.15963581206965683871024158910, 6.79529158808354321457137463369, 7.65939655613263599217977300973, 8.211691624491154162561411547702, 9.565351165630982868470879924864, 10.63085610236776566060054388230