L(s) = 1 | − 1.26i·2-s − 9i·3-s + 30.4·4-s − 11.3·6-s + 49i·7-s − 78.8i·8-s − 81·9-s − 570.·11-s − 273. i·12-s − 512. i·13-s + 61.8·14-s + 873.·16-s + 258. i·17-s + 102. i·18-s + 1.73e3·19-s + ⋯ |
L(s) = 1 | − 0.223i·2-s − 0.577i·3-s + 0.950·4-s − 0.128·6-s + 0.377i·7-s − 0.435i·8-s − 0.333·9-s − 1.42·11-s − 0.548i·12-s − 0.840i·13-s + 0.0843·14-s + 0.852·16-s + 0.216i·17-s + 0.0744i·18-s + 1.10·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5967620365\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5967620365\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49iT \) |
good | 2 | \( 1 + 1.26iT - 32T^{2} \) |
| 11 | \( 1 + 570.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 512. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 258. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.24e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.42e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 7.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.15e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.88e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.79e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 537.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.44e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.09e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.57e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.95e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.45e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.17e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 2.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.82e4iT - 8.58e9T^{2} \) |
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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
−9.795785618717579063127782834328, −8.378675592448081739535199675442, −7.75282776288330810470257723468, −6.89977975839996221781990261447, −5.87004045431032508843347915028, −5.10338772875236954865269357231, −3.24522606035943850415438793014, −2.58216937714481567065289918654, −1.46319605723644883870114278915, −0.11682145747655642540741354530,
1.63706549497140200123825643321, 2.80173860235002391624262228690, 3.81849338167258068405999725386, 5.22483687549925417397460546404, 5.79334828749118768127605011913, 7.34005903536326346644146231127, 7.46571575534961038938109109418, 8.863249950753373868963896965462, 9.799394270446283830060291881676, 10.65874969413955356487607114541