L(s) = 1 | + (0.707 + 0.707i)3-s + (1.79 − 1.32i)5-s + 2.80i·7-s + 1.00i·9-s + (4.12 − 4.12i)11-s + (1.40 − 3.32i)13-s + (2.21 + 0.331i)15-s + (−0.234 − 0.234i)17-s + (−4.70 + 4.70i)19-s + (−1.98 + 1.98i)21-s + (3.02 − 3.02i)23-s + (1.46 − 4.77i)25-s + (−0.707 + 0.707i)27-s + 9.54i·29-s + (4.77 + 4.77i)31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (0.804 − 0.594i)5-s + 1.06i·7-s + 0.333i·9-s + (1.24 − 1.24i)11-s + (0.388 − 0.921i)13-s + (0.570 + 0.0856i)15-s + (−0.0567 − 0.0567i)17-s + (−1.07 + 1.07i)19-s + (−0.433 + 0.433i)21-s + (0.630 − 0.630i)23-s + (0.293 − 0.955i)25-s + (−0.136 + 0.136i)27-s + 1.77i·29-s + (0.857 + 0.857i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.14797 + 0.188377i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.14797 + 0.188377i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.79 + 1.32i)T \) |
| 13 | \( 1 + (-1.40 + 3.32i)T \) |
good | 7 | \( 1 - 2.80iT - 7T^{2} \) |
| 11 | \( 1 + (-4.12 + 4.12i)T - 11iT^{2} \) |
| 17 | \( 1 + (0.234 + 0.234i)T + 17iT^{2} \) |
| 19 | \( 1 + (4.70 - 4.70i)T - 19iT^{2} \) |
| 23 | \( 1 + (-3.02 + 3.02i)T - 23iT^{2} \) |
| 29 | \( 1 - 9.54iT - 29T^{2} \) |
| 31 | \( 1 + (-4.77 - 4.77i)T + 31iT^{2} \) |
| 37 | \( 1 + 8.87iT - 37T^{2} \) |
| 41 | \( 1 + (1.78 + 1.78i)T + 41iT^{2} \) |
| 43 | \( 1 + (2.29 - 2.29i)T - 43iT^{2} \) |
| 47 | \( 1 - 3.25iT - 47T^{2} \) |
| 53 | \( 1 + (4.02 + 4.02i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.98 - 7.98i)T + 59iT^{2} \) |
| 61 | \( 1 + 7.72T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 + (4.50 + 4.50i)T + 71iT^{2} \) |
| 73 | \( 1 + 1.04T + 73T^{2} \) |
| 79 | \( 1 + 7.83iT - 79T^{2} \) |
| 83 | \( 1 - 7.02iT - 83T^{2} \) |
| 89 | \( 1 + (-0.813 - 0.813i)T + 89iT^{2} \) |
| 97 | \( 1 + 6.24T + 97T^{2} \) |
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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
−10.33782218472482417905384672219, −9.142136863424562900391741956463, −8.791196909906535083890772683160, −8.236967929561668636262790372842, −6.55025577013758006334761525176, −5.85409095310341758534831988807, −5.08259746095976846130746586530, −3.77619499240257512899866826321, −2.74487755597851318669593797574, −1.37149515977515868035948617645,
1.41503705446515430762819875740, 2.42389659532418458386005146653, 3.87860890194723581511716050722, 4.62321416484435952966591510914, 6.38664294516684598173903031118, 6.69154161305722746969419923393, 7.46325163520950698313762457289, 8.672514114323954241163567617645, 9.601525321030187686337483090489, 10.00689181342801266167949132255