| L(s) = 1 | + (0.276 + 0.243i)2-s + (0.712 + 1.57i)3-s + (−0.235 − 1.84i)4-s + (−2.72 − 0.395i)5-s + (−0.187 + 0.609i)6-s + (−0.180 − 1.99i)7-s + (0.798 − 1.17i)8-s + (−1.98 + 2.25i)9-s + (−0.655 − 0.771i)10-s + (−2.94 + 2.99i)11-s + (2.75 − 1.68i)12-s + (−5.63 − 0.203i)13-s + (0.434 − 0.593i)14-s + (−1.31 − 4.57i)15-s + (−3.10 + 0.801i)16-s + (3.44 − 1.59i)17-s + ⋯ |
| L(s) = 1 | + (0.195 + 0.172i)2-s + (0.411 + 0.911i)3-s + (−0.117 − 0.924i)4-s + (−1.21 − 0.177i)5-s + (−0.0764 + 0.248i)6-s + (−0.0681 − 0.752i)7-s + (0.282 − 0.416i)8-s + (−0.661 + 0.750i)9-s + (−0.207 − 0.243i)10-s + (−0.888 + 0.904i)11-s + (0.794 − 0.487i)12-s + (−1.56 − 0.0564i)13-s + (0.116 − 0.158i)14-s + (−0.339 − 1.18i)15-s + (−0.776 + 0.200i)16-s + (0.835 − 0.386i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0611035 - 0.207558i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0611035 - 0.207558i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.712 - 1.57i)T \) |
| 59 | \( 1 + (-0.859 + 7.63i)T \) |
| good | 2 | \( 1 + (-0.276 - 0.243i)T + (0.252 + 1.98i)T^{2} \) |
| 5 | \( 1 + (2.72 + 0.395i)T + (4.79 + 1.42i)T^{2} \) |
| 7 | \( 1 + (0.180 + 1.99i)T + (-6.88 + 1.25i)T^{2} \) |
| 11 | \( 1 + (2.94 - 2.99i)T + (-0.198 - 10.9i)T^{2} \) |
| 13 | \( 1 + (5.63 + 0.203i)T + (12.9 + 0.938i)T^{2} \) |
| 17 | \( 1 + (-3.44 + 1.59i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (6.93 + 1.52i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (-4.49 - 1.69i)T + (17.2 + 15.2i)T^{2} \) |
| 29 | \( 1 + (0.186 + 0.927i)T + (-26.7 + 11.2i)T^{2} \) |
| 31 | \( 1 + (0.136 - 0.430i)T + (-25.3 - 17.8i)T^{2} \) |
| 37 | \( 1 + (1.05 + 1.55i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (3.48 + 2.85i)T + (8.08 + 40.1i)T^{2} \) |
| 43 | \( 1 + (-8.19 + 2.11i)T + (37.6 - 20.8i)T^{2} \) |
| 47 | \( 1 + (-7.86 + 1.14i)T + (45.0 - 13.3i)T^{2} \) |
| 53 | \( 1 + (3.21 - 3.78i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (-4.47 - 3.94i)T + (7.68 + 60.5i)T^{2} \) |
| 67 | \( 1 + (5.44 + 11.2i)T + (-41.5 + 52.5i)T^{2} \) |
| 71 | \( 1 + (4.49 - 11.2i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (9.34 + 1.01i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (0.0615 - 3.40i)T + (-78.9 - 2.85i)T^{2} \) |
| 83 | \( 1 + (10.4 - 5.28i)T + (49.0 - 66.9i)T^{2} \) |
| 89 | \( 1 + (0.215 + 0.0724i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (8.65 + 11.8i)T + (-29.3 + 92.4i)T^{2} \) |
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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
−10.38829250913374156765043634176, −9.800095131381091932889486127677, −8.819438611658847280541074406533, −7.59755982627194027640808375006, −7.16252055285053455467940661032, −5.40710658458371786240955714247, −4.64913507066463200237689328633, −4.04979447278588584052418856704, −2.51474050843848999291878077648, −0.10481800349034808545856932742,
2.48608737620251690781672392169, 3.13225442298129314856881367080, 4.32822977758138744974911881004, 5.67818651484705860907665880194, 7.02417211943785312375467830571, 7.74310430129986359361593966456, 8.309067692215216268352063097142, 9.003493584580886123913884044975, 10.57837588660814100126037071855, 11.55373160332838217504395949277
