L(s) = 1 | + (−0.951 − 0.309i)3-s + 3.78i·7-s + (0.809 + 0.587i)9-s + (0.653 − 0.474i)11-s + (−2.79 + 3.84i)13-s + (1.09 − 0.355i)17-s + (−0.00463 − 0.0142i)19-s + (1.17 − 3.60i)21-s + (−3.68 − 5.07i)23-s + (−0.587 − 0.809i)27-s + (−1.14 + 3.51i)29-s + (−0.488 − 1.50i)31-s + (−0.768 + 0.249i)33-s + (−5.02 + 6.91i)37-s + (3.84 − 2.79i)39-s + ⋯ |
L(s) = 1 | + (−0.549 − 0.178i)3-s + 1.43i·7-s + (0.269 + 0.195i)9-s + (0.197 − 0.143i)11-s + (−0.774 + 1.06i)13-s + (0.264 − 0.0861i)17-s + (−0.00106 − 0.00327i)19-s + (0.255 − 0.786i)21-s + (−0.768 − 1.05i)23-s + (−0.113 − 0.155i)27-s + (−0.212 + 0.653i)29-s + (−0.0878 − 0.270i)31-s + (−0.133 + 0.0434i)33-s + (−0.825 + 1.13i)37-s + (0.615 − 0.447i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4827918063\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4827918063\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.78iT - 7T^{2} \) |
| 11 | \( 1 + (-0.653 + 0.474i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.79 - 3.84i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.09 + 0.355i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.00463 + 0.0142i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.68 + 5.07i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.14 - 3.51i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.488 + 1.50i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.02 - 6.91i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (9.30 + 6.75i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + (0.500 + 0.162i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.80 - 0.911i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-9.25 - 6.72i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.54 - 1.84i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (12.6 - 4.10i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.51 - 4.67i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (2.75 + 3.78i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.86 - 8.81i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.35 + 0.439i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (13.0 - 9.46i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-7.66 - 2.49i)T + (78.4 + 57.0i)T^{2} \) |
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
−9.876012384863491773084177274470, −8.878523372618807740742460835753, −8.508524677722857727956529456263, −7.22365946571603649170892758980, −6.60581292865706212233172684715, −5.67080115101902899130477833604, −5.07289605004533180076476936900, −4.00975207353062313722382258007, −2.64188891013527868087853476126, −1.76256685974742672802691672219,
0.20159454457726179180124311876, 1.53419052923012873911252521390, 3.19125545521773202192221191026, 4.04563170823057042901272060424, 4.92668355174603555135165411689, 5.77371577599873369857139638198, 6.75464557589849614427490512653, 7.52880717286403740760175456446, 8.049126895037024728020888134265, 9.395540133222854536709993257838