| L(s) = 1 | − 5-s + 3·7-s + 3·11-s − 8·17-s − 12·19-s − 6·23-s + 5·25-s − 2·29-s + 9·31-s − 3·35-s − 4·37-s − 10·41-s + 6·43-s − 6·47-s + 7·49-s − 26·53-s − 3·55-s + 12·59-s − 8·61-s + 6·67-s + 24·71-s + 18·73-s + 9·77-s + 3·83-s + 8·85-s − 28·89-s + 12·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.904·11-s − 1.94·17-s − 2.75·19-s − 1.25·23-s + 25-s − 0.371·29-s + 1.61·31-s − 0.507·35-s − 0.657·37-s − 1.56·41-s + 0.914·43-s − 0.875·47-s + 49-s − 3.57·53-s − 0.404·55-s + 1.56·59-s − 1.02·61-s + 0.733·67-s + 2.84·71-s + 2.10·73-s + 1.02·77-s + 0.329·83-s + 0.867·85-s − 2.96·89-s + 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9005350153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9005350153\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−9.406722273484293590378104077470, −8.539894729899007531684503809456, −8.235410307686453555642085418454, −7.991985351185401865034700748035, −7.969986391357432693427757482099, −6.80144823076242624029199097355, −6.72321180906767480337033406261, −6.51617517707714741918982648802, −6.34977891971447125115596354407, −5.29807214058478326940552368586, −5.24201393017557749678157021391, −4.51872188632654927870926986462, −4.34575975340095781736265850676, −4.02247775329240768178898620189, −3.63505843256845745132198298754, −2.68745158081229376177402325939, −2.39794959279787705066749597459, −1.75698597975476198504091946700, −1.48030795464861643061788229240, −0.30138793357138054951497464283,
0.30138793357138054951497464283, 1.48030795464861643061788229240, 1.75698597975476198504091946700, 2.39794959279787705066749597459, 2.68745158081229376177402325939, 3.63505843256845745132198298754, 4.02247775329240768178898620189, 4.34575975340095781736265850676, 4.51872188632654927870926986462, 5.24201393017557749678157021391, 5.29807214058478326940552368586, 6.34977891971447125115596354407, 6.51617517707714741918982648802, 6.72321180906767480337033406261, 6.80144823076242624029199097355, 7.969986391357432693427757482099, 7.991985351185401865034700748035, 8.235410307686453555642085418454, 8.539894729899007531684503809456, 9.406722273484293590378104077470
