| L(s) = 1 | − 7.07i·5-s + 60·7-s − 333. i·11-s + 1.19e3·13-s − 4.15e3i·17-s + 8.43e3·19-s − 6.40e3i·23-s + 1.55e4·25-s − 3.36e4i·29-s + 4.68e4·31-s − 424. i·35-s + 1.09e4·37-s − 7.34e4i·41-s + 5.94e4·43-s + 1.17e5i·47-s + ⋯ |
| L(s) = 1 | − 0.0565i·5-s + 0.174·7-s − 0.250i·11-s + 0.542·13-s − 0.844i·17-s + 1.22·19-s − 0.526i·23-s + 0.996·25-s − 1.37i·29-s + 1.57·31-s − 0.00989i·35-s + 0.215·37-s − 1.06i·41-s + 0.747·43-s + 1.13i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.81468 - 0.576775i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81468 - 0.576775i\) |
| \(L(4)\) |
|
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 | \( 1 + 7.07iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 60T + 1.17e5T^{2} \) |
| 11 | \( 1 + 333. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.19e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 4.15e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 8.43e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 6.40e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.36e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 4.68e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 1.09e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 7.34e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 5.94e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.17e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 8.23e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.81e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.39e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.48e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.11e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.01e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.91e4T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.00e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 3.75e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 6.63e5T + 8.32e11T^{2} \) |
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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
−13.45127999915791234107886788597, −12.11691674446768919128572266311, −11.17097249702574704092818816991, −9.898400003545445193927115069038, −8.702034213493401803322214382635, −7.45917952771708308088168974684, −6.03257953984044729760428149218, −4.59108819542034649879879842420, −2.88032563411354170025661491534, −0.891378740307999447886685741437,
1.31007294133183128484313999100, 3.23824999167444395325352189503, 4.87642242505818816693063400240, 6.33536602019541418179959921039, 7.70096539401770132258348756441, 8.923238857139259838921828312510, 10.19797663457264019122350606444, 11.29673655934496479956583614450, 12.43144950501542937261663204720, 13.54405841986054065464433982777
