L(s) = 1 | + (−2.17 − 1.25i)3-s + (−2.95 − 1.70i)5-s + 2.54·7-s + (1.66 + 2.87i)9-s − 1.25i·11-s + (4.22 − 2.44i)13-s + (4.29 + 7.43i)15-s + (3.99 − 6.91i)17-s + (3.96 + 1.81i)19-s + (−5.54 − 3.19i)21-s + (0.355 + 0.615i)23-s + (3.32 + 5.75i)25-s − 0.806i·27-s + (2.45 − 1.41i)29-s − 5.32·31-s + ⋯ |
L(s) = 1 | + (−1.25 − 0.725i)3-s + (−1.32 − 0.763i)5-s + 0.961·7-s + (0.553 + 0.958i)9-s − 0.378i·11-s + (1.17 − 0.677i)13-s + (1.10 + 1.91i)15-s + (0.968 − 1.67i)17-s + (0.908 + 0.416i)19-s + (−1.20 − 0.698i)21-s + (0.0741 + 0.128i)23-s + (0.664 + 1.15i)25-s − 0.155i·27-s + (0.455 − 0.262i)29-s − 0.956·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8576598715\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8576598715\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.96 - 1.81i)T \) |
good | 3 | \( 1 + (2.17 + 1.25i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (2.95 + 1.70i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 2.54T + 7T^{2} \) |
| 11 | \( 1 + 1.25iT - 11T^{2} \) |
| 13 | \( 1 + (-4.22 + 2.44i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.99 + 6.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.355 - 0.615i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.45 + 1.41i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 + 3.20iT - 37T^{2} \) |
| 41 | \( 1 + (-5.26 + 9.11i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.69 - 3.29i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.00 + 6.93i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.905 - 0.522i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.46 - 3.73i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (11.9 - 6.87i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.72 - 1.57i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.34 - 10.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.03 + 5.25i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.17 - 2.04i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 + (-1.39 - 2.40i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.77 - 9.99i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.239400290659438640033551960918, −8.335111167045111382874667102688, −7.62363302943251319107111965452, −7.18102583457373954856102890573, −5.68921863433073224562754215565, −5.40639113010942040065815081429, −4.37591429606601213772053714798, −3.28583318723660258907280094761, −1.24933751427778644507899603290, −0.56768066431474443592116803171,
1.34173864118242339708699581936, 3.34757582387841310197949757653, 4.13426579295482231074458771049, 4.81857667636349808675599696267, 5.87349706712213066560094289584, 6.61277415753897494003841621980, 7.69690158540089337773225874461, 8.208419919690045652421019690604, 9.399110504704246101221164687422, 10.47682531819469828983747794322