L(s) = 1 | + (−0.302 + 1.32i)2-s + (−1.73 − 0.0482i)3-s + (0.131 + 0.0633i)4-s + (1.28 + 3.26i)5-s + (0.588 − 2.28i)6-s + (2.63 + 0.220i)7-s + (−1.82 + 2.28i)8-s + (2.99 + 0.167i)9-s + (−4.72 + 0.712i)10-s + (−0.233 + 0.0720i)11-s + (−0.224 − 0.115i)12-s + (5.88 − 1.81i)13-s + (−1.09 + 3.43i)14-s + (−2.06 − 5.72i)15-s + (−2.29 − 2.88i)16-s + (0.132 − 1.76i)17-s + ⋯ |
L(s) = 1 | + (−0.214 + 0.938i)2-s + (−0.999 − 0.0278i)3-s + (0.0657 + 0.0316i)4-s + (0.573 + 1.46i)5-s + (0.240 − 0.932i)6-s + (0.996 + 0.0832i)7-s + (−0.644 + 0.807i)8-s + (0.998 + 0.0557i)9-s + (−1.49 + 0.225i)10-s + (−0.0703 + 0.0217i)11-s + (−0.0648 − 0.0334i)12-s + (1.63 − 0.503i)13-s + (−0.291 + 0.917i)14-s + (−0.532 − 1.47i)15-s + (−0.574 − 0.720i)16-s + (0.0320 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.407798 + 1.18255i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.407798 + 1.18255i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.73 + 0.0482i)T \) |
| 7 | \( 1 + (-2.63 - 0.220i)T \) |
good | 2 | \( 1 + (0.302 - 1.32i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (-1.28 - 3.26i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (0.233 - 0.0720i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-5.88 + 1.81i)T + (10.7 - 7.32i)T^{2} \) |
| 17 | \( 1 + (-0.132 + 1.76i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-1.79 - 3.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.69 - 2.52i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.455 + 6.07i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + 0.258T + 31T^{2} \) |
| 37 | \( 1 + (6.45 + 4.39i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (1.90 + 0.286i)T + (39.1 + 12.0i)T^{2} \) |
| 43 | \( 1 + (8.64 - 1.30i)T + (41.0 - 12.6i)T^{2} \) |
| 47 | \( 1 + (0.106 - 0.468i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-8.24 + 5.62i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (2.15 + 2.69i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (-13.2 + 6.37i)T + (38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + (-3.92 - 1.88i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.691 - 0.213i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + 1.81T + 79T^{2} \) |
| 83 | \( 1 + (-3.72 - 1.14i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (-7.96 + 7.39i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (5.76 - 9.99i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.37823360005404725570084288419, −10.70320559544663892244549479335, −9.895760433591198483688876065952, −8.388205697581766522981216881551, −7.57560098997503350300178099713, −6.72375324739870183421046181651, −5.96002885110637709460711968617, −5.38278225445964493473739701492, −3.58770171795969953075598012436, −1.94510476893233791965278682728,
1.09278075635644107885866832655, 1.73078580799559560651069157203, 3.92614196168309369789208454564, 4.97024805913439597331542528374, 5.82492295190740622617456830101, 6.82569165708035058423134183297, 8.444469396750429090906992943178, 9.050445366185948386193772341419, 10.20715098844333473380169718333, 10.80616576869211172844134011203