| L(s) = 1 | − 2·2-s + 9·3-s − 28·4-s − 98·5-s − 18·6-s + 240·7-s + 120·8-s + 81·9-s + 196·10-s + 336·11-s − 252·12-s + 342·13-s − 480·14-s − 882·15-s + 656·16-s − 6·17-s − 162·18-s − 361·19-s + 2.74e3·20-s + 2.16e3·21-s − 672·22-s + 2.83e3·23-s + 1.08e3·24-s + 6.47e3·25-s − 684·26-s + 729·27-s − 6.72e3·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.577·3-s − 7/8·4-s − 1.75·5-s − 0.204·6-s + 1.85·7-s + 0.662·8-s + 1/3·9-s + 0.619·10-s + 0.837·11-s − 0.505·12-s + 0.561·13-s − 0.654·14-s − 1.01·15-s + 0.640·16-s − 0.00503·17-s − 0.117·18-s − 0.229·19-s + 1.53·20-s + 1.06·21-s − 0.296·22-s + 1.11·23-s + 0.382·24-s + 2.07·25-s − 0.198·26-s + 0.192·27-s − 1.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.334345761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.334345761\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 19 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 + p T + p^{5} T^{2} \) |
| 5 | \( 1 + 98 T + p^{5} T^{2} \) |
| 7 | \( 1 - 240 T + p^{5} T^{2} \) |
| 11 | \( 1 - 336 T + p^{5} T^{2} \) |
| 13 | \( 1 - 342 T + p^{5} T^{2} \) |
| 17 | \( 1 + 6 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2836 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5902 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2744 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13670 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10990 T + p^{5} T^{2} \) |
| 43 | \( 1 + 4996 T + p^{5} T^{2} \) |
| 47 | \( 1 + 17124 T + p^{5} T^{2} \) |
| 53 | \( 1 + 4470 T + p^{5} T^{2} \) |
| 59 | \( 1 - 26292 T + p^{5} T^{2} \) |
| 61 | \( 1 - 29134 T + p^{5} T^{2} \) |
| 67 | \( 1 + 42052 T + p^{5} T^{2} \) |
| 71 | \( 1 + 26112 T + p^{5} T^{2} \) |
| 73 | \( 1 + 49046 T + p^{5} T^{2} \) |
| 79 | \( 1 - 79056 T + p^{5} T^{2} \) |
| 83 | \( 1 - 9472 T + p^{5} T^{2} \) |
| 89 | \( 1 - 82894 T + p^{5} T^{2} \) |
| 97 | \( 1 - 39850 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.68236812536588713619393565380, −13.16120601196797308680278749577, −11.69496600482320090524374185853, −10.96411592285791878750390509379, −9.029256577003613052442722022834, −8.220371441798017620807143184428, −7.52469752671922012077351601111, −4.71686862579217307184752488671, −3.87236526456255497082805727676, −1.08188733810811448422101517028,
1.08188733810811448422101517028, 3.87236526456255497082805727676, 4.71686862579217307184752488671, 7.52469752671922012077351601111, 8.220371441798017620807143184428, 9.029256577003613052442722022834, 10.96411592285791878750390509379, 11.69496600482320090524374185853, 13.16120601196797308680278749577, 14.68236812536588713619393565380
