| L(s) = 1 | − 5-s + 4·7-s − 3·11-s + 4·13-s + 10·19-s + 6·23-s + 9·29-s − 5·31-s − 4·35-s + 4·37-s + 9·41-s + 10·43-s + 6·47-s + 7·49-s − 24·53-s + 3·55-s − 9·59-s + 10·61-s − 4·65-s − 2·67-s + 6·71-s − 8·73-s − 12·77-s + 4·79-s − 6·83-s − 18·89-s + 16·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.904·11-s + 1.10·13-s + 2.29·19-s + 1.25·23-s + 1.67·29-s − 0.898·31-s − 0.676·35-s + 0.657·37-s + 1.40·41-s + 1.52·43-s + 0.875·47-s + 49-s − 3.29·53-s + 0.404·55-s − 1.17·59-s + 1.28·61-s − 0.496·65-s − 0.244·67-s + 0.712·71-s − 0.936·73-s − 1.36·77-s + 0.450·79-s − 0.658·83-s − 1.90·89-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.385251988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.385251988\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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
−9.509584145005958275689704830079, −9.248839847856189721985601763268, −8.638352095944642759694787610973, −8.364529587754795985425252141992, −8.035605832404872760247455687383, −7.58586718954105188407062594631, −7.21863006699407120818014829366, −7.19097435233571141372816542410, −6.06673908830539985331608599105, −6.02369427964775799546206805766, −5.48974204522978763605888325413, −4.92972423493345658179987439707, −4.65721685080376102509392771859, −4.37140213572430849281427805456, −3.50832141578682170466398750906, −3.17044565313194721402684349304, −2.71347237714298722036415788905, −1.96205772729715479995513566826, −1.14293069379841707352992722486, −0.895295443662450554357463963374,
0.895295443662450554357463963374, 1.14293069379841707352992722486, 1.96205772729715479995513566826, 2.71347237714298722036415788905, 3.17044565313194721402684349304, 3.50832141578682170466398750906, 4.37140213572430849281427805456, 4.65721685080376102509392771859, 4.92972423493345658179987439707, 5.48974204522978763605888325413, 6.02369427964775799546206805766, 6.06673908830539985331608599105, 7.19097435233571141372816542410, 7.21863006699407120818014829366, 7.58586718954105188407062594631, 8.035605832404872760247455687383, 8.364529587754795985425252141992, 8.638352095944642759694787610973, 9.248839847856189721985601763268, 9.509584145005958275689704830079
