| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 12·17-s − 16·23-s − 25-s − 4·27-s + 12·29-s − 3·36-s − 8·43-s − 2·48-s + 14·49-s − 24·51-s − 20·53-s − 4·61-s − 64-s − 12·68-s + 32·69-s + 2·75-s − 32·79-s + 5·81-s − 24·87-s + 16·92-s + 100-s + 4·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s + 2.91·17-s − 3.33·23-s − 1/5·25-s − 0.769·27-s + 2.22·29-s − 1/2·36-s − 1.21·43-s − 0.288·48-s + 2·49-s − 3.36·51-s − 2.74·53-s − 0.512·61-s − 1/8·64-s − 1.45·68-s + 3.85·69-s + 0.230·75-s − 3.60·79-s + 5/9·81-s − 2.57·87-s + 1.66·92-s + 1/10·100-s + 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9651385217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9651385217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.322744553475654579019178608893, −8.072109998916235197105837615063, −7.63143783172067378376054679059, −7.50233812875175255108011869558, −6.89345169608282922067128485984, −6.53871494493932803496032202455, −5.97208297641197493495612767500, −5.90255786825983424374140417080, −5.67281168118077168719861733110, −5.21464337660547116155464114702, −4.60823928175496984421791453168, −4.56725924546548230116991366418, −3.90967077195863446539368182483, −3.73093990139767201189774484145, −3.01022719892157244847149251919, −2.86520048737423861042903124174, −1.85225113094165631182200785856, −1.56538039233603013895554301316, −0.985622654308429601258639247182, −0.34515317747471623257156338591,
0.34515317747471623257156338591, 0.985622654308429601258639247182, 1.56538039233603013895554301316, 1.85225113094165631182200785856, 2.86520048737423861042903124174, 3.01022719892157244847149251919, 3.73093990139767201189774484145, 3.90967077195863446539368182483, 4.56725924546548230116991366418, 4.60823928175496984421791453168, 5.21464337660547116155464114702, 5.67281168118077168719861733110, 5.90255786825983424374140417080, 5.97208297641197493495612767500, 6.53871494493932803496032202455, 6.89345169608282922067128485984, 7.50233812875175255108011869558, 7.63143783172067378376054679059, 8.072109998916235197105837615063, 8.322744553475654579019178608893
