| L(s) = 1 | − 6·5-s + 7-s + 3·17-s + 3·19-s + 17·25-s + 3·31-s − 6·35-s − 7·37-s + 6·41-s − 4·43-s + 3·47-s − 6·49-s + 9·53-s − 3·59-s + 21·61-s − 5·67-s − 21·73-s + 79-s − 12·83-s − 18·85-s − 9·89-s − 18·95-s + 12·97-s + 18·101-s − 27·107-s + 17·109-s − 36·113-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 0.377·7-s + 0.727·17-s + 0.688·19-s + 17/5·25-s + 0.538·31-s − 1.01·35-s − 1.15·37-s + 0.937·41-s − 0.609·43-s + 0.437·47-s − 6/7·49-s + 1.23·53-s − 0.390·59-s + 2.68·61-s − 0.610·67-s − 2.45·73-s + 0.112·79-s − 1.31·83-s − 1.95·85-s − 0.953·89-s − 1.84·95-s + 1.21·97-s + 1.79·101-s − 2.61·107-s + 1.62·109-s − 3.38·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9610669493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9610669493\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 80 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 21 T + 208 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 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
−8.891229789196526699584973053353, −8.802810595207401367227162219915, −8.240193630876150076681895287073, −8.082643426175025286429587676049, −7.75068445031401757509108745163, −7.34414843043702209866063308343, −6.99312791620451371327428818785, −6.85913829725020557560138165666, −6.04412379788499528967370507001, −5.52754792195105726076431462768, −5.28946635012044158854357168084, −4.60627326496824452028773322991, −4.27379625378684572307174116045, −4.04832596896567613260230884064, −3.42240713196593192635160265663, −3.23062914100191225071821560402, −2.67628234950523857077839391646, −1.80569832659804818733369847641, −1.03910547273947240476622048106, −0.41364580359651607146298556711,
0.41364580359651607146298556711, 1.03910547273947240476622048106, 1.80569832659804818733369847641, 2.67628234950523857077839391646, 3.23062914100191225071821560402, 3.42240713196593192635160265663, 4.04832596896567613260230884064, 4.27379625378684572307174116045, 4.60627326496824452028773322991, 5.28946635012044158854357168084, 5.52754792195105726076431462768, 6.04412379788499528967370507001, 6.85913829725020557560138165666, 6.99312791620451371327428818785, 7.34414843043702209866063308343, 7.75068445031401757509108745163, 8.082643426175025286429587676049, 8.240193630876150076681895287073, 8.802810595207401367227162219915, 8.891229789196526699584973053353
