L(s) = 1 | + (−0.642 + 0.766i)4-s + (−0.766 + 0.642i)5-s + (0.366 + 1.36i)7-s + (0.342 + 0.939i)9-s + (−0.173 − 0.984i)16-s + (−1.28 + 0.597i)17-s − i·20-s + (−0.123 − 1.40i)23-s + (0.173 − 0.984i)25-s + (−1.28 − 0.597i)28-s + (−1.15 − 0.811i)35-s + (−0.939 − 0.342i)36-s + (1.40 + 0.123i)43-s + (−0.866 − 0.5i)45-s + (−0.597 + 1.28i)47-s + ⋯ |
L(s) = 1 | + (−0.642 + 0.766i)4-s + (−0.766 + 0.642i)5-s + (0.366 + 1.36i)7-s + (0.342 + 0.939i)9-s + (−0.173 − 0.984i)16-s + (−1.28 + 0.597i)17-s − i·20-s + (−0.123 − 1.40i)23-s + (0.173 − 0.984i)25-s + (−1.28 − 0.597i)28-s + (−1.15 − 0.811i)35-s + (−0.939 − 0.342i)36-s + (1.40 + 0.123i)43-s + (−0.866 − 0.5i)45-s + (−0.597 + 1.28i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6731099790\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6731099790\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.642 - 0.766i)T^{2} \) |
| 3 | \( 1 + (-0.342 - 0.939i)T^{2} \) |
| 7 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.342 - 0.939i)T^{2} \) |
| 17 | \( 1 + (1.28 - 0.597i)T + (0.642 - 0.766i)T^{2} \) |
| 23 | \( 1 + (0.123 + 1.40i)T + (-0.984 + 0.173i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-1.40 - 0.123i)T + (0.984 + 0.173i)T^{2} \) |
| 47 | \( 1 + (0.597 - 1.28i)T + (-0.642 - 0.766i)T^{2} \) |
| 53 | \( 1 + (-0.984 + 0.173i)T^{2} \) |
| 59 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.642 - 0.766i)T^{2} \) |
| 71 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.811 - 1.15i)T + (-0.342 - 0.939i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (0.642 - 0.766i)T^{2} \) |
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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
−9.690700320658930370831366307887, −8.663378704748684819405902489025, −8.403636700280602094215361553520, −7.63054230926474926306961030031, −6.77502498644918294418496675944, −5.76965296068016682937064694560, −4.62252078876531456656121165923, −4.23424579305090711524615864850, −2.88939666291090181038396089750, −2.24014108874784690126845988077,
0.53367304276813458563710460650, 1.54445756601223157737288484673, 3.55216028877336498766659990663, 4.23580961639234187909216113017, 4.76446943483780842279081797165, 5.79346642440716879587197869214, 6.90650403208475704519850871499, 7.44145067060815182399686487604, 8.449145933388672426013932340959, 9.182182832435953302260959176544