L(s) = 1 | + (1.94 − 2.68i)2-s + (0.927 − 2.85i)3-s + (−2.16 − 6.65i)4-s + (−3.89 − 5.36i)5-s + (−5.84 − 8.04i)6-s + (2.47 + 7.60i)7-s + (−9.46 − 3.07i)8-s + (−7.28 − 5.29i)9-s − 22·10-s − 21.0·12-s + (3.23 + 2.35i)13-s + (25.2 + 8.19i)14-s + (−18.9 + 6.14i)15-s + (−4.04 + 2.93i)16-s + (−7.79 − 10.7i)17-s + (−28.3 + 9.22i)18-s + ⋯ |
L(s) = 1 | + (0.974 − 1.34i)2-s + (0.309 − 0.951i)3-s + (−0.540 − 1.66i)4-s + (−0.779 − 1.07i)5-s + (−0.974 − 1.34i)6-s + (0.353 + 1.08i)7-s + (−1.18 − 0.384i)8-s + (−0.809 − 0.587i)9-s − 2.20·10-s − 1.75·12-s + (0.248 + 0.180i)13-s + (1.80 + 0.585i)14-s + (−1.26 + 0.409i)15-s + (−0.252 + 0.183i)16-s + (−0.458 − 0.631i)17-s + (−1.57 + 0.512i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.808129 + 2.31287i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.808129 + 2.31287i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.927 + 2.85i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.94 + 2.68i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (3.89 + 5.36i)T + (-7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (-2.47 - 7.60i)T + (-39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (-3.23 - 2.35i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (7.79 + 10.7i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (-1.85 + 5.70i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + 6.63iT - 529T^{2} \) |
| 29 | \( 1 + (-37.8 + 12.2i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-21.0 - 15.2i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-9.27 - 28.5i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-12.6 - 4.09i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + 42T + 1.84e3T^{2} \) |
| 47 | \( 1 + (82.0 + 26.6i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-35.0 + 48.2i)T + (-868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-63.0 + 20.4i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-9.70 + 7.05i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 + (35.0 + 48.2i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-22.8 - 70.3i)T + (-4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (32.3 + 23.5i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-23.3 - 32.1i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (50.1 + 36.4i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−11.40847978307242292189545721495, −9.835984100587268390790404610191, −8.670198969255423563770542905710, −8.208259799721494067553046506917, −6.60818789717111402306161419510, −5.28439749330804372913582015942, −4.54047235218903439872826938531, −3.15027772003636097619838016139, −2.09523237694257881029268506584, −0.811292809446802322404575139449,
3.18160910696266606170493430349, 3.99304683954337036343731870272, 4.68845270593393339994967509620, 6.01599627571608729456957169680, 7.00232033525939463212403665757, 7.79447689023140537239583808624, 8.504455559902708514102041720316, 10.13534096635930487368673574446, 10.79864977739941511077902996781, 11.68428411747846896595914930907