| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s − 3·16-s + 6·17-s + 6·23-s − 2·25-s + 4·27-s + 36-s + 14·43-s + 6·48-s + 4·49-s − 12·51-s − 18·53-s + 4·61-s − 7·64-s + 6·68-s − 12·69-s + 4·75-s − 2·79-s − 11·81-s + 6·92-s − 2·100-s + 2·103-s − 12·107-s + 4·108-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s − 0.577·12-s − 3/4·16-s + 1.45·17-s + 1.25·23-s − 2/5·25-s + 0.769·27-s + 1/6·36-s + 2.13·43-s + 0.866·48-s + 4/7·49-s − 1.68·51-s − 2.47·53-s + 0.512·61-s − 7/8·64-s + 0.727·68-s − 1.44·69-s + 0.461·75-s − 0.225·79-s − 1.22·81-s + 0.625·92-s − 1/5·100-s + 0.197·103-s − 1.16·107-s + 0.384·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214752811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214752811\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.980886673486165546826452260552, −8.436528347276759765453416330773, −7.86681440859899557940278830497, −7.39002474056279655909055763247, −7.01073579549101950820309060680, −6.48478974329924352959280020551, −5.95303991829253836051682144012, −5.68088478474610308995663356944, −5.05490617752337781949443792469, −4.63514044980194004917391793964, −3.95291048716905267129276524078, −3.12951718765850005954972981453, −2.66425180231818102315444142038, −1.64706713671509607318053571788, −0.73426393234454509309693687056,
0.73426393234454509309693687056, 1.64706713671509607318053571788, 2.66425180231818102315444142038, 3.12951718765850005954972981453, 3.95291048716905267129276524078, 4.63514044980194004917391793964, 5.05490617752337781949443792469, 5.68088478474610308995663356944, 5.95303991829253836051682144012, 6.48478974329924352959280020551, 7.01073579549101950820309060680, 7.39002474056279655909055763247, 7.86681440859899557940278830497, 8.436528347276759765453416330773, 8.980886673486165546826452260552
