| L(s) = 1 | + (−0.520 − 0.577i)2-s + (−0.564 − 0.251i)3-s + (0.145 − 1.38i)4-s + (−0.217 − 0.0462i)5-s + (0.148 + 0.456i)6-s + (−2.13 + 1.55i)8-s + (−1.75 − 1.94i)9-s + (0.0865 + 0.149i)10-s + (−3.14 + 1.05i)11-s + (−0.431 + 0.746i)12-s + (−2.01 + 6.20i)13-s + (0.111 + 0.0808i)15-s + (−0.722 − 0.153i)16-s + (2.90 − 3.22i)17-s + (−0.212 + 2.02i)18-s + (0.304 + 2.89i)19-s + ⋯ |
| L(s) = 1 | + (−0.367 − 0.408i)2-s + (−0.325 − 0.145i)3-s + (0.0729 − 0.693i)4-s + (−0.0973 − 0.0206i)5-s + (0.0606 + 0.186i)6-s + (−0.755 + 0.548i)8-s + (−0.583 − 0.648i)9-s + (0.0273 + 0.0473i)10-s + (−0.948 + 0.317i)11-s + (−0.124 + 0.215i)12-s + (−0.559 + 1.72i)13-s + (0.0287 + 0.0208i)15-s + (−0.180 − 0.0383i)16-s + (0.703 − 0.781i)17-s + (−0.0501 + 0.477i)18-s + (0.0699 + 0.665i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0193875 + 0.0296604i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0193875 + 0.0296604i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (3.14 - 1.05i)T \) |
| good | 2 | \( 1 + (0.520 + 0.577i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (0.564 + 0.251i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (0.217 + 0.0462i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (2.01 - 6.20i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.90 + 3.22i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.304 - 2.89i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.94 + 3.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.05 - 2.21i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (6.73 - 1.43i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (5.16 - 2.29i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-1.08 + 0.786i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + (-0.631 - 6.01i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-1.68 + 0.357i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.996 - 9.48i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (9.41 + 2.00i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.635 + 1.10i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.87 + 8.85i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.584 + 5.55i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (3.03 + 3.36i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (3.48 + 10.7i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-3.96 + 6.86i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.79 + 8.61i)T + (-78.4 - 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05175429526682789041287229476, −10.23174500742314075874694897949, −9.427674944849794767805786558067, −8.749291490754434715789616695522, −7.43974735999580718420101504858, −6.50573141783189570442934959064, −5.56830241141481670745954552065, −4.63865918690794601205699408572, −2.99052700456840663807546139868, −1.71920227950621962189721105844,
0.02271222316587832260579304152, 2.64653888035360339250165104487, 3.56955328960032421162839773260, 5.23102554206983514001617678331, 5.75553539233176567156128860705, 7.21547438389032572499301451730, 7.940489685383907349637785926768, 8.407540328339155538660287209259, 9.681228551396678269009881887306, 10.55560009737993206573646352302
