| L(s) = 1 | + (−1.48 + 1.33i)2-s + (0.414 − 3.97i)4-s + (0.542 − 3.07i)5-s + (2.98 + 3.56i)7-s + (4.71 + 6.46i)8-s + (3.31 + 5.29i)10-s + (2.54 − 0.448i)11-s + (−16.7 + 6.09i)13-s + (−9.20 − 1.29i)14-s + (−15.6 − 3.30i)16-s + (10.3 − 17.9i)17-s + (24.5 − 14.1i)19-s + (−12.0 − 3.43i)20-s + (−3.18 + 4.07i)22-s + (14.6 − 17.4i)23-s + ⋯ |
| L(s) = 1 | + (−0.742 + 0.669i)2-s + (0.103 − 0.994i)4-s + (0.108 − 0.615i)5-s + (0.426 + 0.508i)7-s + (0.588 + 0.808i)8-s + (0.331 + 0.529i)10-s + (0.231 − 0.0408i)11-s + (−1.28 + 0.468i)13-s + (−0.657 − 0.0921i)14-s + (−0.978 − 0.206i)16-s + (0.609 − 1.05i)17-s + (1.29 − 0.744i)19-s + (−0.600 − 0.171i)20-s + (−0.144 + 0.185i)22-s + (0.637 − 0.759i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.17302 - 0.0610036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17302 - 0.0610036i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.48 - 1.33i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.542 + 3.07i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-2.98 - 3.56i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-2.54 + 0.448i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (16.7 - 6.09i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-10.3 + 17.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-24.5 + 14.1i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-14.6 + 17.4i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (5.90 + 2.14i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-22.3 + 26.6i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (22.9 - 39.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-4.27 + 1.55i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-12.6 + 2.23i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-1.37 - 1.64i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 31.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-111. - 19.6i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-63.3 + 53.1i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (10.8 + 29.8i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (0.0648 + 0.0374i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (60.1 + 104. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-0.117 + 0.323i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (32.5 - 89.3i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (59.4 + 102. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-11.2 - 63.5i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.47547427398620364007697559230, −10.06702722058493139788926279077, −9.364719675310334497638355579217, −8.655913009119759075738624761778, −7.55606716696686551718383594605, −6.77910961264882883685840015902, −5.28799566317317289859450556002, −4.85705559755628612817323563412, −2.51329216163662572665891146362, −0.858916016459054234106380371603,
1.24651871485718113216482427466, 2.78463347970029092716902885143, 3.88526798811561041325249791142, 5.38869728312631577599460670903, 7.04692321019857081536008629699, 7.58731947749395965283612600595, 8.650395409318649680693820891430, 9.914846196247642855571706316354, 10.29472036182483328096300905993, 11.27607837730951845733533580021
