| L(s) = 1 | + (0.574 + 2.60i)2-s + (−2.66 + 1.36i)3-s + (−2.84 + 1.31i)4-s + (7.65 + 2.12i)5-s + (−5.09 − 6.17i)6-s + (−6.92 + 6.56i)7-s + (1.40 + 1.84i)8-s + (5.25 − 7.30i)9-s + (−1.14 + 21.1i)10-s + (−7.04 + 5.98i)11-s + (5.78 − 7.39i)12-s + (−6.51 + 2.19i)13-s + (−21.0 − 14.3i)14-s + (−23.3 + 4.79i)15-s + (−12.1 + 14.2i)16-s + (21.5 − 22.7i)17-s + ⋯ |
| L(s) = 1 | + (0.287 + 1.30i)2-s + (−0.889 + 0.455i)3-s + (−0.710 + 0.328i)4-s + (1.53 + 0.425i)5-s + (−0.849 − 1.02i)6-s + (−0.989 + 0.937i)7-s + (0.175 + 0.231i)8-s + (0.584 − 0.811i)9-s + (−0.114 + 2.11i)10-s + (−0.640 + 0.544i)11-s + (0.482 − 0.616i)12-s + (−0.500 + 0.168i)13-s + (−1.50 − 1.02i)14-s + (−1.55 + 0.319i)15-s + (−0.757 + 0.891i)16-s + (1.27 − 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0503i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0359034 - 1.42476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0359034 - 1.42476i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.66 - 1.36i)T \) |
| 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
−13.11408091319403666280879392889, −12.20739103301580543681422526273, −10.78010632279822327504282696963, −9.764006094960123029212302252102, −9.187549898209008538615121888384, −7.20421600738054037812411043932, −6.49250374236270484937573709664, −5.54215465333592796565181815662, −5.06878518797585861220936543074, −2.62487930929610996054633130187,
0.884399692263139777725998449561, 2.22214414898372531992606617504, 3.90031931623148257203157059892, 5.51048386368811651369680611537, 6.27607374279405929899520253424, 7.71726803468090498698946018009, 9.679732146915032162295150365020, 10.25139410677878820295678756116, 10.74730959388533199926898358465, 12.24804486011034768958208153385
