| L(s) = 1 | + (−1.34 − 0.441i)2-s + (1.73 − 0.0302i)3-s + (1.60 + 1.18i)4-s + (−0.721 − 0.860i)5-s + (−2.33 − 0.724i)6-s + (1.31 − 0.479i)7-s + (−1.63 − 2.30i)8-s + (2.99 − 0.104i)9-s + (0.589 + 1.47i)10-s + (1.05 − 1.26i)11-s + (2.82 + 2.00i)12-s + (−0.671 + 0.118i)13-s + (−1.98 + 0.0622i)14-s + (−1.27 − 1.46i)15-s + (1.18 + 3.82i)16-s + (0.907 + 1.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.949 − 0.312i)2-s + (0.999 − 0.0174i)3-s + (0.804 + 0.593i)4-s + (−0.322 − 0.384i)5-s + (−0.955 − 0.295i)6-s + (0.498 − 0.181i)7-s + (−0.579 − 0.815i)8-s + (0.999 − 0.0349i)9-s + (0.186 + 0.466i)10-s + (0.319 − 0.380i)11-s + (0.815 + 0.579i)12-s + (−0.186 + 0.0328i)13-s + (−0.529 + 0.0166i)14-s + (−0.329 − 0.378i)15-s + (0.295 + 0.955i)16-s + (0.220 + 0.381i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07535 - 0.338378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07535 - 0.338378i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 + 0.441i)T \) |
| 3 | \( 1 + (-1.73 + 0.0302i)T \) |
| good | 5 | \( 1 + (0.721 + 0.860i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.31 + 0.479i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.05 + 1.26i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.671 - 0.118i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.907 - 1.57i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.96 - 1.71i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.62 + 1.31i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.92 - 0.691i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (6.21 + 2.26i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.83 + 2.21i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.952 - 5.40i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (7.26 - 8.65i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (11.5 - 4.20i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 6.89iT - 53T^{2} \) |
| 59 | \( 1 + (6.78 + 8.09i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.24 - 3.40i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (11.9 - 2.10i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.80 - 11.7i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.73 + 8.20i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.59 + 9.03i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.29 - 0.228i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (3.52 - 6.10i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.33 + 5.31i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.14546025965272644375942260813, −11.15042972186481421662725281665, −10.03214412415030576046582163724, −9.250900386656741885534963005802, −8.141849245219348943394558237979, −7.82506181928782269570570902997, −6.41312214409831037807751440954, −4.36661709785313892117124720370, −3.10063802012875664982733264063, −1.49920558869759316109100521178,
1.84691220598850628783022992562, 3.32240870083796185829812174095, 5.10589054442774434488223181646, 6.81211494041398387867793564749, 7.55087413373891202213550619237, 8.425151418303016439196741954116, 9.379797282576025705907445944220, 10.14864434981969769536651644357, 11.30291930979233041125860578884, 12.17891110336714584896969872580
