L(s) = 1 | + (−0.586 − 1.62i)3-s − 2.64i·7-s + (−2.31 + 1.91i)9-s + 0.359i·11-s − 4.48i·13-s + 7.99·17-s + (−4.31 + 1.55i)21-s + (4.47 + 2.64i)27-s − 10.7i·29-s + (0.586 − 0.211i)33-s + (−7.31 + 2.63i)39-s − 12.4·47-s − 7.00·49-s + (−4.68 − 13.0i)51-s + (5.05 + 6.11i)63-s + ⋯ |
L(s) = 1 | + (−0.338 − 0.940i)3-s − 0.999i·7-s + (−0.770 + 0.637i)9-s + 0.108i·11-s − 1.24i·13-s + 1.93·17-s + (−0.940 + 0.338i)21-s + (0.860 + 0.509i)27-s − 1.99i·29-s + (0.102 − 0.0367i)33-s + (−1.17 + 0.421i)39-s − 1.81·47-s − 49-s + (−0.656 − 1.82i)51-s + (0.637 + 0.770i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.215666198\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.215666198\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.586 + 1.62i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 11 | \( 1 - 0.359iT - 11T^{2} \) |
| 13 | \( 1 + 4.48iT - 13T^{2} \) |
| 17 | \( 1 - 7.99T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10.7iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 15.0iT - 97T^{2} \) |
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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
−8.388976366598264055956624067590, −7.78643275474152386658688584658, −7.42881838885498760724340548308, −6.37789718027639817311014160820, −5.71182055892951360189273835222, −4.89127065390460481878256829612, −3.66757602569181712572078832711, −2.79016608213254387138747017238, −1.41222132808456929696604379093, −0.48082497055179577479366881988,
1.52332908405175634207205167007, 2.95526484758851228212125263782, 3.63709164529683102791959750102, 4.78939100476911651659480966370, 5.37076155331754255036257902569, 6.12655684537884869497831239354, 6.97991471889603071628968370435, 8.126480505069569547677692510520, 8.843191742290513102195119005313, 9.472715662264211653674573600521