| L(s) = 1 | − 6·2-s + 27·3-s − 92·4-s − 162·6-s + 64·7-s + 1.32e3·8-s + 729·9-s − 948·11-s − 2.48e3·12-s + 5.09e3·13-s − 384·14-s + 3.85e3·16-s − 2.83e4·17-s − 4.37e3·18-s − 8.62e3·19-s + 1.72e3·21-s + 5.68e3·22-s + 1.52e4·23-s + 3.56e4·24-s − 3.05e4·26-s + 1.96e4·27-s − 5.88e3·28-s + 3.65e4·29-s − 2.76e5·31-s − 1.92e5·32-s − 2.55e4·33-s + 1.70e5·34-s + ⋯ |
| L(s) = 1 | − 0.530·2-s + 0.577·3-s − 0.718·4-s − 0.306·6-s + 0.0705·7-s + 0.911·8-s + 1/3·9-s − 0.214·11-s − 0.414·12-s + 0.643·13-s − 0.0374·14-s + 0.235·16-s − 1.40·17-s − 0.176·18-s − 0.288·19-s + 0.0407·21-s + 0.113·22-s + 0.262·23-s + 0.526·24-s − 0.341·26-s + 0.192·27-s − 0.0506·28-s + 0.277·29-s − 1.66·31-s − 1.03·32-s − 0.123·33-s + 0.743·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 64 T + p^{7} T^{2} \) |
| 11 | \( 1 + 948 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5098 T + p^{7} T^{2} \) |
| 17 | \( 1 + 28386 T + p^{7} T^{2} \) |
| 19 | \( 1 + 8620 T + p^{7} T^{2} \) |
| 23 | \( 1 - 15288 T + p^{7} T^{2} \) |
| 29 | \( 1 - 36510 T + p^{7} T^{2} \) |
| 31 | \( 1 + 276808 T + p^{7} T^{2} \) |
| 37 | \( 1 + 268526 T + p^{7} T^{2} \) |
| 41 | \( 1 + 629718 T + p^{7} T^{2} \) |
| 43 | \( 1 + 685772 T + p^{7} T^{2} \) |
| 47 | \( 1 + 583296 T + p^{7} T^{2} \) |
| 53 | \( 1 - 428058 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1306380 T + p^{7} T^{2} \) |
| 61 | \( 1 - 300662 T + p^{7} T^{2} \) |
| 67 | \( 1 - 507244 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5560632 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1369082 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6913720 T + p^{7} T^{2} \) |
| 83 | \( 1 - 4376748 T + p^{7} T^{2} \) |
| 89 | \( 1 + 8528310 T + p^{7} T^{2} \) |
| 97 | \( 1 - 8826814 T + p^{7} T^{2} \) |
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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
−12.90899629170101606814239506455, −11.18709112718531932496653723205, −10.08900940400237195845216841166, −8.929231150691246109000272832028, −8.288927942706428154565646572769, −6.87122362662039265666322735215, −4.99887208551786807683805672884, −3.66033208967544635713432022352, −1.75070598369303286956411496560, 0,
1.75070598369303286956411496560, 3.66033208967544635713432022352, 4.99887208551786807683805672884, 6.87122362662039265666322735215, 8.288927942706428154565646572769, 8.929231150691246109000272832028, 10.08900940400237195845216841166, 11.18709112718531932496653723205, 12.90899629170101606814239506455
