| L(s) = 1 | − i·5-s − 7-s + 3i·11-s + 5.29i·13-s − 5.29·17-s + 5.29i·19-s + 5.29·23-s + 4·25-s − 6i·29-s + 7·31-s + i·35-s + 5.29i·37-s + 5.29·41-s + 10.5i·43-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 0.377·7-s + 0.904i·11-s + 1.46i·13-s − 1.28·17-s + 1.21i·19-s + 1.10·23-s + 0.800·25-s − 1.11i·29-s + 1.25·31-s + 0.169i·35-s + 0.869i·37-s + 0.826·41-s + 1.61i·43-s − 0.857·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03024 + 0.711981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03024 + 0.711981i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 13 | \( 1 - 5.29iT - 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 5.29iT - 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 - 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 7iT - 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−10.17244455792205321579181996667, −9.433286064886499730734725991034, −8.803396234530108032524415797020, −7.79483439297028115085737682214, −6.76447490344419391691729859587, −6.19556225715919085845076537437, −4.69068992595319934355912885743, −4.30791330210656127218419276463, −2.75533440554800242906182538349, −1.50478418137638362998695337896,
0.63138255125390564134012319601, 2.66421434806051776112425496936, 3.29749953629428359706880374856, 4.70751325147152446347623099662, 5.63874328248793152202329051402, 6.65388656503007674644501821836, 7.26937519188725795931641992834, 8.539644032329400118984286783300, 8.962116895421446649655230786697, 10.16742462637024965429564743434
