| L(s) = 1 | − 2·5-s + 6·7-s − 12·11-s + 13-s − 2·17-s − 8·19-s + 5·25-s − 2·29-s − 2·31-s − 12·35-s − 14·37-s + 43-s + 13·49-s − 4·53-s + 24·55-s − 8·59-s + 11·61-s − 2·65-s − 15·67-s − 6·71-s − 9·73-s − 72·77-s + 13·79-s − 28·83-s + 4·85-s + 12·89-s + 6·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2.26·7-s − 3.61·11-s + 0.277·13-s − 0.485·17-s − 1.83·19-s + 25-s − 0.371·29-s − 0.359·31-s − 2.02·35-s − 2.30·37-s + 0.152·43-s + 13/7·49-s − 0.549·53-s + 3.23·55-s − 1.04·59-s + 1.40·61-s − 0.248·65-s − 1.83·67-s − 0.712·71-s − 1.05·73-s − 8.20·77-s + 1.46·79-s − 3.07·83-s + 0.433·85-s + 1.27·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4348214768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4348214768\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 158 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.34068756620246985831017438481, −8.998168709704336046560122042400, −8.954002627747658894026416721498, −8.311716297648978780958570229556, −8.135164510251059153697413825935, −7.88285798135470726235766528744, −7.67231064504793062558735913754, −6.97343054171246210416975473379, −6.81217165422627643114563796790, −5.70990634921058613077608227380, −5.62754883012658754571120704636, −4.94321129437007001119596786741, −4.91136873809083980733563698111, −4.38393869178348063718615890204, −3.96110463130137320156465563036, −2.94877821156996220038676751647, −2.81031159691183865612239614066, −1.83160629286675538816495346508, −1.82054907416013604360406302184, −0.25270772619315161914226936669,
0.25270772619315161914226936669, 1.82054907416013604360406302184, 1.83160629286675538816495346508, 2.81031159691183865612239614066, 2.94877821156996220038676751647, 3.96110463130137320156465563036, 4.38393869178348063718615890204, 4.91136873809083980733563698111, 4.94321129437007001119596786741, 5.62754883012658754571120704636, 5.70990634921058613077608227380, 6.81217165422627643114563796790, 6.97343054171246210416975473379, 7.67231064504793062558735913754, 7.88285798135470726235766528744, 8.135164510251059153697413825935, 8.311716297648978780958570229556, 8.954002627747658894026416721498, 8.998168709704336046560122042400, 10.34068756620246985831017438481
