| 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
−10.63085610236776566060054388230, −9.565351165630982868470879924864, −8.211691624491154162561411547702, −7.65939655613263599217977300973, −6.79529158808354321457137463369, −6.15963581206965683871024158910, −5.10682264309828942400842904404, −2.96487915006744858357021861503, −2.35594546444084911528761148237, −0.70244799795944258119014251231,
2.16402511441150797442928135567, 3.72240456151987258086946270619, 4.50790816033526812573190738198, 5.44657680420239610876292575015, 6.06972711139620324632373984255, 7.891246890773051475159617723420, 8.991600323536996309795660354084, 9.230164618047807929472459195072, 10.19665192361274187024739870039, 10.83580734695944948155423385135
