L(s) = 1 | + 0.424i·2-s + (−0.625 − 1.61i)3-s + 1.82·4-s + (−1.80 + 3.12i)5-s + (0.685 − 0.265i)6-s + 1.62i·8-s + (−2.21 + 2.01i)9-s + (−1.32 − 0.765i)10-s + (−3.20 + 1.85i)11-s + (−1.13 − 2.94i)12-s + (−5.23 + 3.02i)13-s + (6.17 + 0.960i)15-s + 2.95·16-s + (−0.532 + 0.921i)17-s + (−0.856 − 0.941i)18-s + (3.16 − 1.82i)19-s + ⋯ |
L(s) = 1 | + 0.299i·2-s + (−0.360 − 0.932i)3-s + 0.910·4-s + (−0.806 + 1.39i)5-s + (0.279 − 0.108i)6-s + 0.572i·8-s + (−0.739 + 0.673i)9-s + (−0.419 − 0.241i)10-s + (−0.967 + 0.558i)11-s + (−0.328 − 0.848i)12-s + (−1.45 + 0.838i)13-s + (1.59 + 0.248i)15-s + 0.738·16-s + (−0.129 + 0.223i)17-s + (−0.201 − 0.221i)18-s + (0.725 − 0.418i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.485872 + 0.729301i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.485872 + 0.729301i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.625 + 1.61i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.424iT - 2T^{2} \) |
| 5 | \( 1 + (1.80 - 3.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.20 - 1.85i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.23 - 3.02i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.532 - 0.921i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.16 + 1.82i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.314 + 0.181i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.857 + 0.495i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.08iT - 31T^{2} \) |
| 37 | \( 1 + (-4.00 - 6.93i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.09 - 3.62i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.89 - 3.28i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 5.67T + 47T^{2} \) |
| 53 | \( 1 + (3.92 + 2.26i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 0.0275iT - 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 + 5.55iT - 71T^{2} \) |
| 73 | \( 1 + (-1.95 - 1.12i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 + (1.52 - 2.64i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.47 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.67 + 0.964i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.55692216537803195454438914520, −10.76496625367717646180064540514, −9.885019805149377447914795763270, −8.071044035573392561214173157126, −7.38313278101912209519437537894, −7.03718854339632944496237680952, −6.14308154114174251774287444613, −4.84702681019419292166967601174, −2.94695104519880589816011755144, −2.22958085872103353606069309227,
0.53069524395666459630267615679, 2.74424419878555630472677777073, 3.91690684777071122887942576283, 5.11989907960607426728160170245, 5.65365483188879088097128591424, 7.37271647819285283426020248992, 8.084507629618525116362441143951, 9.191364770585878692124774051727, 10.09948443853348746999108250420, 10.85446761343453747739196530774