| L(s) = 1 | + (−0.998 + 0.0627i)2-s + (−0.904 + 0.425i)3-s + (0.992 − 0.125i)4-s + (−0.729 + 2.11i)5-s + (0.876 − 0.481i)6-s + (3.02 + 4.16i)7-s + (−0.982 + 0.187i)8-s + (0.637 − 0.770i)9-s + (0.595 − 2.15i)10-s + (0.399 + 6.35i)11-s + (−0.844 + 0.535i)12-s + (−1.38 − 1.14i)13-s + (−3.27 − 3.96i)14-s + (−0.240 − 2.22i)15-s + (0.968 − 0.248i)16-s + (−0.106 + 0.840i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 + 0.0443i)2-s + (−0.522 + 0.245i)3-s + (0.496 − 0.0626i)4-s + (−0.326 + 0.945i)5-s + (0.357 − 0.196i)6-s + (1.14 + 1.57i)7-s + (−0.347 + 0.0662i)8-s + (0.212 − 0.256i)9-s + (0.188 − 0.681i)10-s + (0.120 + 1.91i)11-s + (−0.243 + 0.154i)12-s + (−0.383 − 0.317i)13-s + (−0.876 − 1.05i)14-s + (−0.0620 − 0.574i)15-s + (0.242 − 0.0621i)16-s + (−0.0257 + 0.203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.262335 + 0.837518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.262335 + 0.837518i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.998 - 0.0627i)T \) |
| 3 | \( 1 + (0.904 - 0.425i)T \) |
| 5 | \( 1 + (0.729 - 2.11i)T \) |
| good | 7 | \( 1 + (-3.02 - 4.16i)T + (-2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.399 - 6.35i)T + (-10.9 + 1.37i)T^{2} \) |
| 13 | \( 1 + (1.38 + 1.14i)T + (2.43 + 12.7i)T^{2} \) |
| 17 | \( 1 + (0.106 - 0.840i)T + (-16.4 - 4.22i)T^{2} \) |
| 19 | \( 1 + (-2.70 + 5.75i)T + (-12.1 - 14.6i)T^{2} \) |
| 23 | \( 1 + (-2.22 - 5.63i)T + (-16.7 + 15.7i)T^{2} \) |
| 29 | \( 1 + (-6.28 + 5.90i)T + (1.82 - 28.9i)T^{2} \) |
| 31 | \( 1 + (4.25 + 0.537i)T + (30.0 + 7.70i)T^{2} \) |
| 37 | \( 1 + (-0.179 - 0.698i)T + (-32.4 + 17.8i)T^{2} \) |
| 41 | \( 1 + (3.57 + 1.41i)T + (29.8 + 28.0i)T^{2} \) |
| 43 | \( 1 + (-9.02 - 2.93i)T + (34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (6.50 + 1.24i)T + (43.6 + 17.3i)T^{2} \) |
| 53 | \( 1 + (4.35 - 7.92i)T + (-28.3 - 44.7i)T^{2} \) |
| 59 | \( 1 + (6.45 + 10.1i)T + (-25.1 + 53.3i)T^{2} \) |
| 61 | \( 1 + (-4.25 + 1.68i)T + (44.4 - 41.7i)T^{2} \) |
| 67 | \( 1 + (-0.0743 + 0.0791i)T + (-4.20 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-0.601 + 3.15i)T + (-66.0 - 26.1i)T^{2} \) |
| 73 | \( 1 + (1.85 + 1.17i)T + (31.0 + 66.0i)T^{2} \) |
| 79 | \( 1 + (-2.51 - 5.35i)T + (-50.3 + 60.8i)T^{2} \) |
| 83 | \( 1 + (-5.68 - 2.67i)T + (52.9 + 63.9i)T^{2} \) |
| 89 | \( 1 + (-1.44 + 2.26i)T + (-37.8 - 80.5i)T^{2} \) |
| 97 | \( 1 + (-1.47 - 1.56i)T + (-6.09 + 96.8i)T^{2} \) |
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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
−10.76661096206919900073754223299, −9.696869819352182504807318689809, −9.254379454175513569995582837898, −7.977529139754950508581505889285, −7.38247160435462638043700416238, −6.45713774577612098974775051596, −5.33208983639943154023573498679, −4.57767217816367574482428573351, −2.80850027687320931280646463997, −1.88228397935612522107751048043,
0.66702526985665781627317648468, 1.40848117275349425453402895822, 3.51078236613158820633258772801, 4.59350373962242545600921831251, 5.51111161159025958446450247098, 6.69569015449589918191465872453, 7.64945140847089240448493175796, 8.235518210628704643545861729392, 8.939645604123445345217843742141, 10.24245602611576451081005628568
