| L(s) = 1 | − 3-s + 3·5-s − 7-s + 11-s + 8·13-s − 3·15-s − 4·17-s + 21-s + 8·23-s + 5·25-s + 27-s + 14·29-s + 11·31-s − 33-s − 3·35-s + 4·37-s − 8·39-s − 8·41-s − 4·43-s − 2·47-s − 6·49-s + 4·51-s − 11·53-s + 3·55-s − 7·59-s + 10·61-s + 24·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.377·7-s + 0.301·11-s + 2.21·13-s − 0.774·15-s − 0.970·17-s + 0.218·21-s + 1.66·23-s + 25-s + 0.192·27-s + 2.59·29-s + 1.97·31-s − 0.174·33-s − 0.507·35-s + 0.657·37-s − 1.28·39-s − 1.24·41-s − 0.609·43-s − 0.291·47-s − 6/7·49-s + 0.560·51-s − 1.51·53-s + 0.404·55-s − 0.911·59-s + 1.28·61-s + 2.97·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.009175092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.009175092\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 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
−10.06403949339596612909995604869, −9.186874160606983069984929416596, −9.125591191316717170756914902042, −8.739257275064680281031669429951, −8.143548307247605719678740202827, −8.129923352657131114395774051163, −7.13039351053344942575014776823, −6.57117942376118690062142446769, −6.37990816668717623083536430168, −6.32655495123957651176649343127, −5.92032540992724889093630859078, −5.06163116513150962783386038976, −4.84034314503276473901531131539, −4.53249032800872432696424561415, −3.66771491238042644943726516818, −3.10046686024821182551724976966, −2.83685834219060416299373919959, −1.98184731264477549097822052644, −1.26248597761430183616232730659, −0.873195450882738156505419366204,
0.873195450882738156505419366204, 1.26248597761430183616232730659, 1.98184731264477549097822052644, 2.83685834219060416299373919959, 3.10046686024821182551724976966, 3.66771491238042644943726516818, 4.53249032800872432696424561415, 4.84034314503276473901531131539, 5.06163116513150962783386038976, 5.92032540992724889093630859078, 6.32655495123957651176649343127, 6.37990816668717623083536430168, 6.57117942376118690062142446769, 7.13039351053344942575014776823, 8.129923352657131114395774051163, 8.143548307247605719678740202827, 8.739257275064680281031669429951, 9.125591191316717170756914902042, 9.186874160606983069984929416596, 10.06403949339596612909995604869
