| L(s) = 1 | + 9.97i·5-s + 2.20·7-s − 9·9-s − 20.3i·11-s + 18.9·17-s − 19i·19-s − 34.8·23-s − 74.4·25-s + 22.0i·35-s − 53.8i·43-s − 89.7i·45-s + 36.6·47-s − 44.1·49-s + 203.·55-s − 121. i·61-s + ⋯ |
| L(s) = 1 | + 1.99i·5-s + 0.315·7-s − 9-s − 1.85i·11-s + 1.11·17-s − i·19-s − 1.51·23-s − 2.97·25-s + 0.629i·35-s − 1.25i·43-s − 1.99i·45-s + 0.779·47-s − 0.900·49-s + 3.69·55-s − 1.99i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9377971146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9377971146\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19iT \) |
| good | 3 | \( 1 + 9T^{2} \) |
| 5 | \( 1 - 9.97iT - 25T^{2} \) |
| 7 | \( 1 - 2.20T + 49T^{2} \) |
| 11 | \( 1 + 20.3iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 18.9T + 289T^{2} \) |
| 23 | \( 1 + 34.8T + 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 53.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 36.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 121. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 112.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 90iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.473244677180240948035791841798, −8.322174107673224452387239679575, −7.83651579766777467597625861554, −6.76830033811753895395873157711, −6.04789262852512059812110682163, −5.47375231051640143728803903500, −3.68482589359772244903796971184, −3.16313575748285918876117213038, −2.29021377464043412122419509728, −0.28302573297101250628464372836,
1.23395014084039058477960692535, 2.12563377435366652091667137808, 3.86143581756980540012405886923, 4.64788236766440068991420885238, 5.37793225128400268564948982156, 6.08419935739664782692267628964, 7.70711941315072372215607272856, 7.996705537079329767245281295495, 8.890596216836830042938977155032, 9.675650277960248703463040286839
