L(s) = 1 | + (−0.258 + 0.965i)2-s + (0.382 − 0.923i)3-s + (−0.866 − 0.499i)4-s + (0.991 − 0.130i)5-s + (0.793 + 0.608i)6-s + (−0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.130 + 0.991i)10-s + (−0.793 + 0.608i)12-s + (−1.78 + 0.478i)13-s + (0.499 − 0.866i)14-s + (0.258 − 0.965i)15-s + (0.500 + 0.866i)16-s + (0.866 − 0.5i)18-s − 1.21i·19-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (0.382 − 0.923i)3-s + (−0.866 − 0.499i)4-s + (0.991 − 0.130i)5-s + (0.793 + 0.608i)6-s + (−0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.130 + 0.991i)10-s + (−0.793 + 0.608i)12-s + (−1.78 + 0.478i)13-s + (0.499 − 0.866i)14-s + (0.258 − 0.965i)15-s + (0.500 + 0.866i)16-s + (0.866 − 0.5i)18-s − 1.21i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0136 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0136 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7805232553\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7805232553\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 + (-0.382 + 0.923i)T \) |
| 5 | \( 1 + (-0.991 + 0.130i)T \) |
| 7 | \( 1 + (0.965 + 0.258i)T \) |
good | 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + 1.21iT - T^{2} \) |
| 23 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.793 + 1.37i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 0.517iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.739 + 0.198i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
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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
−8.909695271048251667413765006678, −8.071228561103548002978608329912, −7.10741518879779404927267391452, −6.74387595997402705127770362120, −6.17467144158346245484262216576, −5.18883513709230761521261904112, −4.38122788284718933425996312981, −2.88742361180613641029364733650, −2.08471572462920194314624214709, −0.49872759441500041889746850457,
1.85805046834335967777862208495, 2.73904446284260932504089168903, 3.34228764563113711071341280250, 4.31932987493344971432496316061, 5.39062981873899923706577655311, 5.74961336802512939741659326822, 7.22209617359082344696969778479, 7.976475075946023173772391398018, 8.990465859273031585944698957470, 9.525101427038448596070987640651