| L(s) = 1 | − 3·5-s − 7-s − 3·11-s + 4·13-s − 4·19-s − 6·23-s + 5·25-s − 6·29-s + 5·31-s + 3·35-s + 4·37-s + 6·41-s − 10·43-s + 6·47-s + 7·49-s + 18·53-s + 9·55-s + 12·59-s − 8·61-s − 12·65-s + 14·67-s − 14·73-s + 3·77-s + 8·79-s − 3·83-s − 36·89-s − 4·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 0.904·11-s + 1.10·13-s − 0.917·19-s − 1.25·23-s + 25-s − 1.11·29-s + 0.898·31-s + 0.507·35-s + 0.657·37-s + 0.937·41-s − 1.52·43-s + 0.875·47-s + 49-s + 2.47·53-s + 1.21·55-s + 1.56·59-s − 1.02·61-s − 1.48·65-s + 1.71·67-s − 1.63·73-s + 0.341·77-s + 0.900·79-s − 0.329·83-s − 3.81·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7964814620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7964814620\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.746266926549570975963666163121, −9.656006873732270803153995633813, −8.740725989049939507801918146223, −8.521399002859435344255609083733, −8.413971585554655254693153641808, −7.83609555994059738411538301002, −7.44804446916367992049576119955, −7.11695985635625060933175497586, −6.58559162243052681813754138401, −6.15837862759726484282065971427, −5.55335235845374044294954377329, −5.46345401361076513034899102011, −4.54736471386251336464197687542, −4.09898884746470333322458933721, −3.92096703262733784497332058272, −3.44322270379388536072764451070, −2.60322991193936342984643976652, −2.36182743128603628280398755613, −1.30144393276342706793242850071, −0.39594265362025642370081711560,
0.39594265362025642370081711560, 1.30144393276342706793242850071, 2.36182743128603628280398755613, 2.60322991193936342984643976652, 3.44322270379388536072764451070, 3.92096703262733784497332058272, 4.09898884746470333322458933721, 4.54736471386251336464197687542, 5.46345401361076513034899102011, 5.55335235845374044294954377329, 6.15837862759726484282065971427, 6.58559162243052681813754138401, 7.11695985635625060933175497586, 7.44804446916367992049576119955, 7.83609555994059738411538301002, 8.413971585554655254693153641808, 8.521399002859435344255609083733, 8.740725989049939507801918146223, 9.656006873732270803153995633813, 9.746266926549570975963666163121
