L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (1.08 + 0.216i)5-s + (0.923 + 0.382i)8-s + (0.382 − 0.923i)9-s + (−0.216 − 1.08i)10-s + (1 − i)13-s − i·16-s + (−0.382 − 0.923i)17-s − 18-s + (−0.923 + 0.617i)20-s + (0.216 + 0.0897i)25-s + (−1.30 − 0.541i)26-s + (−1.63 − 0.324i)29-s + (−0.923 + 0.382i)32-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (1.08 + 0.216i)5-s + (0.923 + 0.382i)8-s + (0.382 − 0.923i)9-s + (−0.216 − 1.08i)10-s + (1 − i)13-s − i·16-s + (−0.382 − 0.923i)17-s − 18-s + (−0.923 + 0.617i)20-s + (0.216 + 0.0897i)25-s + (−1.30 − 0.541i)26-s + (−1.63 − 0.324i)29-s + (−0.923 + 0.382i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.227661670\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.227661670\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 + 0.923i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.382 + 0.923i)T \) |
good | 3 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 5 | \( 1 + (-1.08 - 0.216i)T + (0.923 + 0.382i)T^{2} \) |
| 11 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 29 | \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (-0.617 - 0.923i)T + (-0.382 + 0.923i)T^{2} \) |
| 41 | \( 1 + (-0.382 + 0.0761i)T + (0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \) |
| 79 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 97 | \( 1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2} \) |
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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
−8.987183303320061800956201414169, −7.996412376623181727365598802433, −7.24884497719123227907869547385, −6.24079212784694546754272915972, −5.64509022129563463739862496777, −4.59872219267987249139177859934, −3.64220783691536117804935198785, −2.91933398413382598881907258715, −1.94429569771089455934424791282, −0.909859101426476994500068359206,
1.52519391527340899580865129749, 2.05971571441448430910075645070, 3.86580807294393818615755398860, 4.53294544040660625011824602240, 5.60871292832768869792417810431, 5.88501089487096725604937127772, 6.79367709706006810692297921905, 7.46042468403722050466587981776, 8.330783998170380150620377610558, 8.980068915202811715868581538401