| L(s) = 1 | + 4·2-s − 66·5-s − 176·7-s − 64·8-s − 264·10-s − 60·11-s + 658·13-s − 704·14-s − 256·16-s + 828·17-s + 1.91e3·19-s − 240·22-s + 600·23-s + 3.12e3·25-s + 2.63e3·26-s + 5.57e3·29-s + 3.59e3·31-s + 3.31e3·34-s + 1.16e4·35-s − 1.69e4·37-s + 7.64e3·38-s + 4.22e3·40-s + 1.91e4·41-s − 1.33e4·43-s + 2.40e3·46-s − 1.96e4·47-s + 1.68e4·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.18·5-s − 1.35·7-s − 0.353·8-s − 0.834·10-s − 0.149·11-s + 1.07·13-s − 0.959·14-s − 1/4·16-s + 0.694·17-s + 1.21·19-s − 0.105·22-s + 0.236·23-s + 25-s + 0.763·26-s + 1.23·29-s + 0.671·31-s + 0.491·34-s + 1.60·35-s − 2.03·37-s + 0.859·38-s + 0.417·40-s + 1.78·41-s − 1.09·43-s + 0.167·46-s − 1.29·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.093724146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.093724146\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 66 T + 1231 T^{2} + 66 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 176 T + 14169 T^{2} + 176 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 60 T - 157451 T^{2} + 60 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 658 T + 61671 T^{2} - 658 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 414 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 956 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 600 T - 6076343 T^{2} - 600 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5574 T + 10558327 T^{2} - 5574 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3592 T - 15726687 T^{2} - 3592 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8458 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 19194 T + 252553435 T^{2} - 19194 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13316 T + 30307413 T^{2} + 13316 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 19680 T + 157957393 T^{2} + 19680 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 31266 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 26340 T - 21128699 T^{2} - 26340 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 31090 T + 121991799 T^{2} - 31090 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 16804 T - 1067750691 T^{2} - 16804 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6120 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 25558 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 74408 T + 2459494065 T^{2} + 74408 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6468 T - 3897205619 T^{2} + 6468 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 32742 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 166082 T + 18995890467 T^{2} + 166082 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.03862313276269817443236552260, −11.98875818054334684521571466455, −11.44599180217576130406331988773, −10.75131981621552018410509488516, −10.14450077903738874267086658423, −9.864808419257020752482010093121, −9.066480256802874639223227844858, −8.529197023935716213472784550273, −8.183547698941227998986595933953, −7.33626192498415116717211737082, −6.85952592027834732659941452985, −6.43993901506786695393637258436, −5.50639799335371246321401698355, −5.27988647598606773773556913638, −4.17053614048430523978843520804, −3.83726253818949210828191344914, −3.15687136856707435364339014561, −2.79572615860706246305250595709, −1.18268506728603258127753662652, −0.49019946487129206942234873435,
0.49019946487129206942234873435, 1.18268506728603258127753662652, 2.79572615860706246305250595709, 3.15687136856707435364339014561, 3.83726253818949210828191344914, 4.17053614048430523978843520804, 5.27988647598606773773556913638, 5.50639799335371246321401698355, 6.43993901506786695393637258436, 6.85952592027834732659941452985, 7.33626192498415116717211737082, 8.183547698941227998986595933953, 8.529197023935716213472784550273, 9.066480256802874639223227844858, 9.864808419257020752482010093121, 10.14450077903738874267086658423, 10.75131981621552018410509488516, 11.44599180217576130406331988773, 11.98875818054334684521571466455, 12.03862313276269817443236552260
