| L(s) = 1 | − 2·2-s + 3·4-s + 5-s − 5·7-s − 4·8-s − 2·10-s + 5·11-s + 10·14-s + 5·16-s − 4·17-s − 8·19-s + 3·20-s − 10·22-s − 4·23-s + 5·25-s − 15·28-s − 5·29-s + 6·31-s − 6·32-s + 8·34-s − 5·35-s + 4·37-s + 16·38-s − 4·40-s − 2·43-s + 15·44-s + 8·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.447·5-s − 1.88·7-s − 1.41·8-s − 0.632·10-s + 1.50·11-s + 2.67·14-s + 5/4·16-s − 0.970·17-s − 1.83·19-s + 0.670·20-s − 2.13·22-s − 0.834·23-s + 25-s − 2.83·28-s − 0.928·29-s + 1.07·31-s − 1.06·32-s + 1.37·34-s − 0.845·35-s + 0.657·37-s + 2.59·38-s − 0.632·40-s − 0.304·43-s + 2.26·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5436923444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5436923444\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7 T - 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 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
−9.840462188827719874446867679209, −9.649022718894528597203314728578, −9.212879486606263588610389039542, −8.766752875597712008886375625666, −8.679389745117110926165653873899, −8.246795330553539931853325491137, −7.40192486737818341742223158679, −7.04444614829709831000873990967, −6.83237166736305553226023105055, −6.21493109140247131124802962384, −6.09797370327935468571856435792, −5.88960415092835783548164444614, −4.78199494654529479242810511242, −4.15497241411548340926350688589, −3.85728875618723341816388641216, −3.15630869334358871749293131531, −2.52381867671227407277573328748, −2.15700104508673027605128167644, −1.32990217166099898527693036394, −0.42027559159147366791037994264,
0.42027559159147366791037994264, 1.32990217166099898527693036394, 2.15700104508673027605128167644, 2.52381867671227407277573328748, 3.15630869334358871749293131531, 3.85728875618723341816388641216, 4.15497241411548340926350688589, 4.78199494654529479242810511242, 5.88960415092835783548164444614, 6.09797370327935468571856435792, 6.21493109140247131124802962384, 6.83237166736305553226023105055, 7.04444614829709831000873990967, 7.40192486737818341742223158679, 8.246795330553539931853325491137, 8.679389745117110926165653873899, 8.766752875597712008886375625666, 9.212879486606263588610389039542, 9.649022718894528597203314728578, 9.840462188827719874446867679209
