| L(s) = 1 | + 86·3-s + 5.20e3·9-s + 1.19e5·19-s − 1.56e5·25-s + 2.59e5·27-s + 4.41e5·43-s + 1.64e6·49-s + 1.02e7·57-s + 7.70e6·67-s − 9.73e6·73-s − 1.34e7·75-s + 1.09e7·81-s − 1.98e7·97-s + 3.87e7·121-s + 127-s + 3.79e7·129-s + 131-s + 137-s + 139-s + 1.41e8·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.25e8·169-s + 6.22e8·171-s + ⋯ |
| L(s) = 1 | + 1.83·3-s + 2.38·9-s + 3.99·19-s − 2·25-s + 2.54·27-s + 0.845·43-s + 2·49-s + 7.34·57-s + 3.12·67-s − 2.92·73-s − 3.67·75-s + 2.29·81-s − 2.21·97-s + 1.98·121-s + 1.55·129-s + 3.67·147-s + 2·169-s + 9.51·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.641943843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.641943843\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 86 T + p^{7} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 8814 T + p^{7} T^{2} )( 1 + 8814 T + p^{7} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 22182 T + p^{7} T^{2} )( 1 + 22182 T + p^{7} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 59722 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 236886 T + p^{7} T^{2} )( 1 + 236886 T + p^{7} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 220510 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 1030926 T + p^{7} T^{2} )( 1 + 1030926 T + p^{7} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3851302 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4865614 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4808934 T + p^{7} T^{2} )( 1 + 4808934 T + p^{7} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 7073118 T + p^{7} T^{2} )( 1 + 7073118 T + p^{7} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 9938890 T + p^{7} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.99449384294397823486740535855, −12.40521427991133248951895605458, −11.67582665394903434112873330848, −11.51091923953674624927079521535, −10.37549027327442447000107705022, −9.850424350189971014298817668960, −9.503948452633549792331625147036, −9.149584671733048675197414690681, −8.355914242485764156439537423376, −7.77386915414209514424318906856, −7.41914627081992842191158660226, −7.01532292066109958335551457852, −5.74512879443780058446754487664, −5.30408542862336064351619599249, −4.20966893059618951853820231846, −3.64508122409479828632137077039, −3.05069767868799247848694848553, −2.41080442021850776080061162411, −1.49007162154777631293915785325, −0.834436916877825013230349285082,
0.834436916877825013230349285082, 1.49007162154777631293915785325, 2.41080442021850776080061162411, 3.05069767868799247848694848553, 3.64508122409479828632137077039, 4.20966893059618951853820231846, 5.30408542862336064351619599249, 5.74512879443780058446754487664, 7.01532292066109958335551457852, 7.41914627081992842191158660226, 7.77386915414209514424318906856, 8.355914242485764156439537423376, 9.149584671733048675197414690681, 9.503948452633549792331625147036, 9.850424350189971014298817668960, 10.37549027327442447000107705022, 11.51091923953674624927079521535, 11.67582665394903434112873330848, 12.40521427991133248951895605458, 12.99449384294397823486740535855
