Error: no document with id 270871356 found in table mf_hecke_traces.
Error: table True does not exist
| L(s) = 1 | + 1.84·2-s + 1.39·4-s + (−0.667 + 1.15i)5-s − 1.12·8-s + (−1.22 + 2.12i)10-s + (0.756 + 1.31i)11-s + (2.58 + 4.48i)13-s − 4.84·16-s + (0.774 − 1.34i)17-s + (1.25 + 2.16i)19-s + (−0.927 + 1.60i)20-s + (1.39 + 2.41i)22-s + (−3.68 + 6.37i)23-s + (1.60 + 2.78i)25-s + (4.76 + 8.25i)26-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.695·4-s + (−0.298 + 0.516i)5-s − 0.396·8-s + (−0.388 + 0.673i)10-s + (0.228 + 0.395i)11-s + (0.717 + 1.24i)13-s − 1.21·16-s + (0.187 − 0.325i)17-s + (0.287 + 0.497i)19-s + (−0.207 + 0.359i)20-s + (0.296 + 0.514i)22-s + (−0.767 + 1.32i)23-s + (0.321 + 0.557i)25-s + (0.934 + 1.61i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.658394559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.658394559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.84T + 2T^{2} \) |
| 5 | \( 1 + (0.667 - 1.15i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.756 - 1.31i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.58 - 4.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.774 + 1.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 2.16i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.68 - 6.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0309 + 0.0536i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.51 - 7.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.09 + 8.83i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 + (0.755 - 1.30i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 + 3.23T + 61T^{2} \) |
| 67 | \( 1 - 6.93T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (-1.37 + 2.38i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + (-2.80 + 4.85i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.703 - 1.21i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.09 + 10.5i)T + (-48.5 - 84.0i)T^{2} \) |
| 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
−9.716143231115287024318488548582, −9.166758366159932276170100877658, −8.011522104327768930153920223458, −7.07246077425080348325543312608, −6.37017774794438543251331601927, −5.58052949096504849407966454260, −4.57195287755826674309291029601, −3.82591783916503433755947554713, −3.09841296048539237113105413520, −1.75472250703718710996755910379,
0.72962784186715403578149165869, 2.59196752696090038464740569431, 3.49879582973067264842779433142, 4.34677452643856105038174837853, 5.09819334139767424879615520597, 6.00154784680940492330648626434, 6.54923118244629450151733084053, 7.951136714284638836037677493805, 8.471838058297990796614727892172, 9.384478417980635023442185650623
