L(s) = 1 | − 2-s + (1.57 − 0.721i)3-s + 4-s + (−1.57 + 0.721i)6-s + (2.29 − 1.31i)7-s − 8-s + (1.95 − 2.27i)9-s − 3.91i·11-s + (1.57 − 0.721i)12-s − 4.99·13-s + (−2.29 + 1.31i)14-s + 16-s − 3.54i·17-s + (−1.95 + 2.27i)18-s − 3.14i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.909 − 0.416i)3-s + 0.5·4-s + (−0.642 + 0.294i)6-s + (0.867 − 0.496i)7-s − 0.353·8-s + (0.652 − 0.757i)9-s − 1.18i·11-s + (0.454 − 0.208i)12-s − 1.38·13-s + (−0.613 + 0.351i)14-s + 0.250·16-s − 0.860i·17-s + (−0.461 + 0.535i)18-s − 0.722i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.503216204\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.503216204\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.57 + 0.721i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.29 + 1.31i)T \) |
good | 11 | \( 1 + 3.91iT - 11T^{2} \) |
| 13 | \( 1 + 4.99T + 13T^{2} \) |
| 17 | \( 1 + 3.54iT - 17T^{2} \) |
| 19 | \( 1 + 3.14iT - 19T^{2} \) |
| 23 | \( 1 + 7.54T + 23T^{2} \) |
| 29 | \( 1 - 7.54iT - 29T^{2} \) |
| 31 | \( 1 - 4.19iT - 31T^{2} \) |
| 37 | \( 1 + 10.4iT - 37T^{2} \) |
| 41 | \( 1 - 9.32T + 41T^{2} \) |
| 43 | \( 1 - 2.91iT - 43T^{2} \) |
| 47 | \( 1 - 8.00iT - 47T^{2} \) |
| 53 | \( 1 - 0.288T + 53T^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 - 2.48iT - 61T^{2} \) |
| 67 | \( 1 - 0.545iT - 67T^{2} \) |
| 71 | \( 1 + 5.37iT - 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 + 0.742T + 79T^{2} \) |
| 83 | \( 1 - 4.45iT - 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 0.524T + 97T^{2} \) |
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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
−9.417469457560602263970294155073, −8.899754170786984345041595597925, −7.930490314012272326905532602321, −7.50845886069160548901438419237, −6.70123500556437583339774582699, −5.42930019938295597814760755968, −4.25992345020096340317654560311, −3.02738604300038167598835813228, −2.09733954835952850192402001619, −0.74124230016518951995884508653,
1.92963280569703110967307646040, 2.36156316716072542746271582608, 3.97859672972082429143978418211, 4.75835716918532365186295260572, 5.91603357586970604093892125699, 7.20974010879660370198037364413, 7.977012929437703673374737939703, 8.286855860255704671660657279572, 9.522904924241416667415040397521, 9.878566047373466865457781009559