L(s) = 1 | + 4·4-s + 12·16-s − 16·19-s + 22·31-s − 2·49-s + 28·61-s + 32·64-s − 64·76-s + 26·79-s − 34·109-s − 22·121-s + 88·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 2·4-s + 3·16-s − 3.67·19-s + 3.95·31-s − 2/7·49-s + 3.58·61-s + 4·64-s − 7.34·76-s + 2.92·79-s − 3.25·109-s − 2·121-s + 7.90·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.202113755\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.202113755\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−10.75819168588572343848323113666, −10.54858495810987629510598853463, −9.844529891341369787784231284664, −9.816811288976397281213416712894, −8.774919480841514488881111832692, −8.246968342162871053097042095081, −8.243589701591840791104326560879, −7.77945312641817069444077985275, −6.82165287914696189416544050321, −6.74221228843304988180411145042, −6.37155950846698374631844754781, −6.15021755989021311848870605310, −5.38504436553706873675352535559, −4.75183781769020535089670801982, −4.11623386204961117031660887611, −3.67043398966982035742569579679, −2.60420127112829702617233401114, −2.53993077091533218404738570485, −1.95430030547790610867132311866, −0.959655429978204059344517526566,
0.959655429978204059344517526566, 1.95430030547790610867132311866, 2.53993077091533218404738570485, 2.60420127112829702617233401114, 3.67043398966982035742569579679, 4.11623386204961117031660887611, 4.75183781769020535089670801982, 5.38504436553706873675352535559, 6.15021755989021311848870605310, 6.37155950846698374631844754781, 6.74221228843304988180411145042, 6.82165287914696189416544050321, 7.77945312641817069444077985275, 8.243589701591840791104326560879, 8.246968342162871053097042095081, 8.774919480841514488881111832692, 9.816811288976397281213416712894, 9.844529891341369787784231284664, 10.54858495810987629510598853463, 10.75819168588572343848323113666