| L(s) = 1 | + (−0.574 − 2.60i)2-s + (2.09 + 2.14i)3-s + (−2.84 + 1.31i)4-s + (−7.65 − 2.12i)5-s + (4.40 − 6.69i)6-s + (−6.92 + 6.56i)7-s + (−1.40 − 1.84i)8-s + (−0.235 + 8.99i)9-s + (−1.14 + 21.1i)10-s + (7.04 − 5.98i)11-s + (−8.77 − 3.35i)12-s + (−6.51 + 2.19i)13-s + (21.0 + 14.3i)14-s + (−11.4 − 20.9i)15-s + (−12.1 + 14.2i)16-s + (−21.5 + 22.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.287 − 1.30i)2-s + (0.697 + 0.716i)3-s + (−0.710 + 0.328i)4-s + (−1.53 − 0.425i)5-s + (0.733 − 1.11i)6-s + (−0.989 + 0.937i)7-s + (−0.175 − 0.231i)8-s + (−0.0261 + 0.999i)9-s + (−0.114 + 2.11i)10-s + (0.640 − 0.544i)11-s + (−0.730 − 0.279i)12-s + (−0.500 + 0.168i)13-s + (1.50 + 1.02i)14-s + (−0.764 − 1.39i)15-s + (−0.757 + 0.891i)16-s + (−1.27 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0642171 + 0.0917293i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0642171 + 0.0917293i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.09 - 2.14i)T \) |
| 59 | \( 1 + (-37.5 + 45.5i)T \) |
| good | 2 | \( 1 + (0.574 + 2.60i)T + (-3.63 + 1.67i)T^{2} \) |
| 5 | \( 1 + (7.65 + 2.12i)T + (21.4 + 12.8i)T^{2} \) |
| 7 | \( 1 + (6.92 - 6.56i)T + (2.65 - 48.9i)T^{2} \) |
| 11 | \( 1 + (-7.04 + 5.98i)T + (19.5 - 119. i)T^{2} \) |
| 13 | \( 1 + (6.51 - 2.19i)T + (134. - 102. i)T^{2} \) |
| 17 | \( 1 + (21.5 - 22.7i)T + (-15.6 - 288. i)T^{2} \) |
| 19 | \( 1 + (7.52 + 18.8i)T + (-262. + 248. i)T^{2} \) |
| 23 | \( 1 + (0.0328 + 0.301i)T + (-516. + 113. i)T^{2} \) |
| 29 | \( 1 + (-8.26 + 37.5i)T + (-763. - 353. i)T^{2} \) |
| 31 | \( 1 + (0.793 - 1.99i)T + (-697. - 660. i)T^{2} \) |
| 37 | \( 1 + (-20.2 - 15.4i)T + (366. + 1.31e3i)T^{2} \) |
| 41 | \( 1 + (-3.54 + 32.5i)T + (-1.64e3 - 361. i)T^{2} \) |
| 43 | \( 1 + (-11.4 + 13.4i)T + (-299. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (82.6 - 22.9i)T + (1.89e3 - 1.13e3i)T^{2} \) |
| 53 | \( 1 + (20.5 - 1.11i)T + (2.79e3 - 303. i)T^{2} \) |
| 61 | \( 1 + (-5.51 + 1.21i)T + (3.37e3 - 1.56e3i)T^{2} \) |
| 67 | \( 1 + (-71.1 + 54.0i)T + (1.20e3 - 4.32e3i)T^{2} \) |
| 71 | \( 1 + (120. - 33.3i)T + (4.31e3 - 2.59e3i)T^{2} \) |
| 73 | \( 1 + (16.2 - 23.9i)T + (-1.97e3 - 4.95e3i)T^{2} \) |
| 79 | \( 1 + (-11.7 - 71.5i)T + (-5.91e3 + 1.99e3i)T^{2} \) |
| 83 | \( 1 + (18.6 - 9.88i)T + (3.86e3 - 5.70e3i)T^{2} \) |
| 89 | \( 1 + (0.251 - 1.14i)T + (-7.18e3 - 3.32e3i)T^{2} \) |
| 97 | \( 1 + (-41.3 - 60.9i)T + (-3.48e3 + 8.74e3i)T^{2} \) |
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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
−12.54825280294708442800022784156, −11.58586784286702941797487063003, −10.97025606224266722318169718174, −9.708335728380795984255011771611, −8.916879970623627325814275187260, −8.297860802326555104940467206238, −6.51019954331772384797663622280, −4.40904051213948691284416125473, −3.60772648964471539445712449872, −2.46801590267889812692555576374,
0.06398549626311981391838204980, 3.07491138023881400439781258720, 4.34265651491215994597253650728, 6.56403166112086278333925875245, 7.05654528301493737889628510559, 7.65579867044739859972708645050, 8.683150188348633141557572297005, 9.749110420407117238715249136106, 11.35684419232296345169401433359, 12.27001071736616878739356778490
