| L(s) = 1 | − 6·5-s − 5·9-s − 8·13-s + 12·17-s + 17·25-s − 2·37-s + 30·45-s − 10·49-s − 12·53-s − 8·61-s + 48·65-s − 8·73-s + 16·81-s − 72·85-s − 18·89-s − 14·97-s + 36·101-s + 4·109-s − 30·113-s + 40·117-s + 121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 5/3·9-s − 2.21·13-s + 2.91·17-s + 17/5·25-s − 0.328·37-s + 4.47·45-s − 1.42·49-s − 1.64·53-s − 1.02·61-s + 5.95·65-s − 0.936·73-s + 16/9·81-s − 7.80·85-s − 1.90·89-s − 1.42·97-s + 3.58·101-s + 0.383·109-s − 2.82·113-s + 3.69·117-s + 1/11·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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
−11.52932662806962264853358891658, −11.33372769095583858463760631653, −10.42805632650209012086523138057, −9.813753565180213813561671136388, −9.146174464181464239622953038476, −8.132665807481574488057539990847, −7.992911069953808459961425963145, −7.60852684015633966980596452192, −6.99346836799162715894329128020, −5.78719635391273558621520357914, −5.11227132195889762323685903220, −4.40400371694164943841525727402, −3.27381985666804460223319703873, −3.10139317797780286714650046035, 0,
3.10139317797780286714650046035, 3.27381985666804460223319703873, 4.40400371694164943841525727402, 5.11227132195889762323685903220, 5.78719635391273558621520357914, 6.99346836799162715894329128020, 7.60852684015633966980596452192, 7.992911069953808459961425963145, 8.132665807481574488057539990847, 9.146174464181464239622953038476, 9.813753565180213813561671136388, 10.42805632650209012086523138057, 11.33372769095583858463760631653, 11.52932662806962264853358891658
