| L(s) = 1 | − 2-s + i·3-s + 4-s − i·5-s − i·6-s − 2i·7-s − 8-s + 2·9-s + i·10-s + i·12-s − 13-s + 2i·14-s + 15-s + 16-s − 2·18-s + 19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 0.755i·7-s − 0.353·8-s + 0.666·9-s + 0.316i·10-s + 0.288i·12-s − 0.277·13-s + 0.534i·14-s + 0.258·15-s + 0.250·16-s − 0.471·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.357461296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.357461296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 3iT - 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 11iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 9iT - 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−8.971807693348313794363075309246, −8.041946526838945207586179364508, −7.38918945195630894883718980601, −6.79405755285783134557960016009, −5.69217033429252380983012910551, −4.85047623749186750355832344846, −4.05255765866557071944721068145, −3.23604327551875863742923757466, −1.84356532577172858367452623102, −0.850117469981375646397209454533,
0.77642438952377509808926225928, 2.07323033654967975724996696862, 2.62948967064601736084175058998, 3.87617957209786499444344952136, 4.94531090693676494515945538013, 6.04991708267063342041788012685, 6.50324233649328710401482021501, 7.49396520703997928083938559322, 7.74876565077571323941013902205, 8.856828113652526361111622126523
