| L(s) = 1 | + 2·5-s + 6·7-s + 2·9-s + 2·11-s − 4·13-s + 2·17-s − 6·19-s + 2·23-s − 25-s + 14·29-s + 12·35-s − 12·37-s − 8·43-s + 4·45-s − 14·47-s + 18·49-s + 4·55-s + 6·59-s − 2·61-s + 12·63-s − 8·65-s − 8·67-s − 6·73-s + 12·77-s + 16·79-s − 5·81-s + 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.26·7-s + 2/3·9-s + 0.603·11-s − 1.10·13-s + 0.485·17-s − 1.37·19-s + 0.417·23-s − 1/5·25-s + 2.59·29-s + 2.02·35-s − 1.97·37-s − 1.21·43-s + 0.596·45-s − 2.04·47-s + 18/7·49-s + 0.539·55-s + 0.781·59-s − 0.256·61-s + 1.51·63-s − 0.992·65-s − 0.977·67-s − 0.702·73-s + 1.36·77-s + 1.80·79-s − 5/9·81-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.431901276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.431901276\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 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
−11.81853148565904024045506306947, −11.49478479548073334636179847494, −10.85972639865483693286913601197, −10.44146808139289874391322627518, −10.03828948999061258209431281768, −9.753678489553911571767561693619, −8.885835115585751563821797233951, −8.570992365075063202755649707221, −8.123015815808663174689436146854, −7.71631869699299992646644216015, −6.83672439412938143272034418773, −6.76307353317260986551227096086, −5.95958106222581037048951865545, −5.18088978233152992064949977803, −4.70574979518464936507486512757, −4.68421099951534354003189335228, −3.66745985351084340687865058748, −2.61823717490455430736711609819, −1.80819253553579409390043188520, −1.44839225881263002157215345488,
1.44839225881263002157215345488, 1.80819253553579409390043188520, 2.61823717490455430736711609819, 3.66745985351084340687865058748, 4.68421099951534354003189335228, 4.70574979518464936507486512757, 5.18088978233152992064949977803, 5.95958106222581037048951865545, 6.76307353317260986551227096086, 6.83672439412938143272034418773, 7.71631869699299992646644216015, 8.123015815808663174689436146854, 8.570992365075063202755649707221, 8.885835115585751563821797233951, 9.753678489553911571767561693619, 10.03828948999061258209431281768, 10.44146808139289874391322627518, 10.85972639865483693286913601197, 11.49478479548073334636179847494, 11.81853148565904024045506306947
