L(s) = 1 | − 0.305·2-s + (1.47 − 0.913i)3-s − 1.90·4-s + 2.05i·5-s + (−0.450 + 0.279i)6-s + (0.946 + 2.47i)7-s + 1.19·8-s + (1.33 − 2.68i)9-s − 0.628i·10-s + 5.03·11-s + (−2.80 + 1.74i)12-s + (−1.75 + 3.14i)13-s + (−0.289 − 0.755i)14-s + (1.87 + 3.02i)15-s + 3.44·16-s + 3.24·17-s + ⋯ |
L(s) = 1 | − 0.216·2-s + (0.849 − 0.527i)3-s − 0.953·4-s + 0.918i·5-s + (−0.183 + 0.114i)6-s + (0.357 + 0.933i)7-s + 0.422·8-s + (0.444 − 0.895i)9-s − 0.198i·10-s + 1.51·11-s + (−0.810 + 0.502i)12-s + (−0.487 + 0.872i)13-s + (−0.0774 − 0.201i)14-s + (0.484 + 0.780i)15-s + 0.861·16-s + 0.787·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30356 + 0.272176i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30356 + 0.272176i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.47 + 0.913i)T \) |
| 7 | \( 1 + (-0.946 - 2.47i)T \) |
| 13 | \( 1 + (1.75 - 3.14i)T \) |
good | 2 | \( 1 + 0.305T + 2T^{2} \) |
| 5 | \( 1 - 2.05iT - 5T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 - 5.12iT - 29T^{2} \) |
| 31 | \( 1 + 5.11T + 31T^{2} \) |
| 37 | \( 1 + 4.85iT - 37T^{2} \) |
| 41 | \( 1 - 3.35iT - 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 - 7.14iT - 47T^{2} \) |
| 53 | \( 1 + 3.08iT - 53T^{2} \) |
| 59 | \( 1 + 12.5iT - 59T^{2} \) |
| 61 | \( 1 + 3.48iT - 61T^{2} \) |
| 67 | \( 1 + 7.77iT - 67T^{2} \) |
| 71 | \( 1 - 6.61T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 + 0.936iT - 83T^{2} \) |
| 89 | \( 1 + 11.5iT - 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−12.21612772417612939350436777462, −11.01477551366918581874679454609, −9.692229376952075279401254975954, −9.030466014035028554518850106592, −8.342154263104684513402913853607, −7.11859067443341614730513599631, −6.23127333622633074791753111572, −4.53142907694113064183335848697, −3.33954419835574957622590429696, −1.83387199489709858795907139800,
1.27421592979404916715604316513, 3.67364185608639823208106403814, 4.36587112683805047398735042387, 5.39056813039958593476110523353, 7.35076943858940118125837586965, 8.239954939096954626799273832052, 8.977497321884474656060031570266, 9.764129040172913610428522341292, 10.50250131010860464465199640915, 11.94041544484754692655382743959