| L(s) = 1 | − 4·5-s + 8·7-s + 8·11-s + 4·13-s + 2·17-s − 8·23-s + 11·25-s − 4·31-s − 32·35-s + 4·37-s + 14·41-s − 4·43-s − 16·47-s + 34·49-s − 2·53-s − 32·55-s + 16·61-s − 16·65-s − 4·71-s + 64·77-s + 24·83-s − 8·85-s + 14·89-s + 32·91-s − 8·103-s + 20·107-s + 14·109-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 3.02·7-s + 2.41·11-s + 1.10·13-s + 0.485·17-s − 1.66·23-s + 11/5·25-s − 0.718·31-s − 5.40·35-s + 0.657·37-s + 2.18·41-s − 0.609·43-s − 2.33·47-s + 34/7·49-s − 0.274·53-s − 4.31·55-s + 2.04·61-s − 1.98·65-s − 0.474·71-s + 7.29·77-s + 2.63·83-s − 0.867·85-s + 1.48·89-s + 3.35·91-s − 0.788·103-s + 1.93·107-s + 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.010907091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.010907091\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.864186566902324167306207206356, −8.702485728305937122240165124564, −8.222069823722670893258557749798, −8.161715169508802989485601516354, −7.65105355146711739553546518657, −7.59905239547221614206630681754, −6.92104240342536557308562116937, −6.57235071275820981555627261467, −6.00786522236643714865551178674, −5.72623267185225693359926744426, −4.93455600730360021723529990105, −4.74668014663121018950039784247, −4.20615008684324163419850536930, −4.12766402297140582703333299584, −3.54330036235743708146218811587, −3.36222366196629802242552391027, −2.02031145430530760498514150503, −1.91233755490504936093116145918, −1.12028807030612311359358263882, −0.873067245960450676732979338121,
0.873067245960450676732979338121, 1.12028807030612311359358263882, 1.91233755490504936093116145918, 2.02031145430530760498514150503, 3.36222366196629802242552391027, 3.54330036235743708146218811587, 4.12766402297140582703333299584, 4.20615008684324163419850536930, 4.74668014663121018950039784247, 4.93455600730360021723529990105, 5.72623267185225693359926744426, 6.00786522236643714865551178674, 6.57235071275820981555627261467, 6.92104240342536557308562116937, 7.59905239547221614206630681754, 7.65105355146711739553546518657, 8.161715169508802989485601516354, 8.222069823722670893258557749798, 8.702485728305937122240165124564, 8.864186566902324167306207206356
