| L(s) = 1 | − 2·5-s − 6·7-s + 12·11-s + 13-s − 2·17-s + 8·19-s + 5·25-s − 2·29-s + 2·31-s + 12·35-s − 14·37-s − 43-s + 13·49-s − 4·53-s − 24·55-s + 8·59-s + 11·61-s − 2·65-s + 15·67-s + 6·71-s − 9·73-s − 72·77-s − 13·79-s + 28·83-s + 4·85-s + 12·89-s − 6·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2.26·7-s + 3.61·11-s + 0.277·13-s − 0.485·17-s + 1.83·19-s + 25-s − 0.371·29-s + 0.359·31-s + 2.02·35-s − 2.30·37-s − 0.152·43-s + 13/7·49-s − 0.549·53-s − 3.23·55-s + 1.04·59-s + 1.40·61-s − 0.248·65-s + 1.83·67-s + 0.712·71-s − 1.05·73-s − 8.20·77-s − 1.46·79-s + 3.07·83-s + 0.433·85-s + 1.27·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.756232022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.756232022\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 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
−9.166364681294549300852532771614, −8.815751935529598399336026629237, −8.343902814544177251246791840366, −7.929373110075289372246179410398, −7.21891069205695543263464810000, −6.94953495799220625386005164377, −6.74235916654225749158340644987, −6.46676353902859107088307218351, −6.28225212430359841448048758840, −5.55342501522582746160853348767, −5.18886732773353441794442939639, −4.59947917396128513003517154896, −3.79026423806617108421397104815, −3.74982591028905006728197135366, −3.68330849469412559022062920112, −3.14966209925132612810778793289, −2.55189422188455684418291350211, −1.59536403484124578836069784750, −1.15352962241163900241119319056, −0.50175226794472828149829641733,
0.50175226794472828149829641733, 1.15352962241163900241119319056, 1.59536403484124578836069784750, 2.55189422188455684418291350211, 3.14966209925132612810778793289, 3.68330849469412559022062920112, 3.74982591028905006728197135366, 3.79026423806617108421397104815, 4.59947917396128513003517154896, 5.18886732773353441794442939639, 5.55342501522582746160853348767, 6.28225212430359841448048758840, 6.46676353902859107088307218351, 6.74235916654225749158340644987, 6.94953495799220625386005164377, 7.21891069205695543263464810000, 7.929373110075289372246179410398, 8.343902814544177251246791840366, 8.815751935529598399336026629237, 9.166364681294549300852532771614
