| L(s) = 1 | + 4·3-s + 8·9-s + 12·13-s − 16-s + 8·25-s + 12·27-s + 4·31-s − 12·37-s + 48·39-s − 4·48-s + 32·61-s + 28·67-s − 8·73-s + 32·75-s + 23·81-s + 16·93-s + 44·97-s − 24·103-s − 48·111-s + 96·117-s + 40·121-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 149-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s + 3.32·13-s − 1/4·16-s + 8/5·25-s + 2.30·27-s + 0.718·31-s − 1.97·37-s + 7.68·39-s − 0.577·48-s + 4.09·61-s + 3.42·67-s − 0.936·73-s + 3.69·75-s + 23/9·81-s + 1.65·93-s + 4.46·97-s − 2.36·103-s − 4.55·111-s + 8.87·117-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2/3·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.22235663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.22235663\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 3682 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 4786 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 3122 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.26891475208656824449279522470, −7.11885553515330570538620492867, −6.76147903473644709712080002943, −6.60118200056126273517061255496, −6.41930049700097427792334693327, −6.17944588651546687327925668722, −5.82998894297209056577540208367, −5.78827381665689402033656916842, −5.17688625951214733387887437133, −5.15553263580872571377333604639, −4.77768891981928012535306282329, −4.73485587314305410390142953187, −4.07264686419896078577388497806, −3.95117509700163715706981941832, −3.75142164105122242155079255180, −3.49016883387830525602920195176, −3.35248100317936595055224117306, −3.25524980856768513062933791808, −2.70313833160141553854625577755, −2.42141718024908486135479049505, −2.16753496969179128945682982213, −1.95917235941421539842317946104, −1.39013957921369484586671644610, −0.954639368283717911635974826442, −0.888377897893704031347493990203,
0.888377897893704031347493990203, 0.954639368283717911635974826442, 1.39013957921369484586671644610, 1.95917235941421539842317946104, 2.16753496969179128945682982213, 2.42141718024908486135479049505, 2.70313833160141553854625577755, 3.25524980856768513062933791808, 3.35248100317936595055224117306, 3.49016883387830525602920195176, 3.75142164105122242155079255180, 3.95117509700163715706981941832, 4.07264686419896078577388497806, 4.73485587314305410390142953187, 4.77768891981928012535306282329, 5.15553263580872571377333604639, 5.17688625951214733387887437133, 5.78827381665689402033656916842, 5.82998894297209056577540208367, 6.17944588651546687327925668722, 6.41930049700097427792334693327, 6.60118200056126273517061255496, 6.76147903473644709712080002943, 7.11885553515330570538620492867, 7.26891475208656824449279522470
