| L(s) = 1 | − 2·7-s + 2·19-s + 25-s + 31-s − 37-s + 3·49-s + 3·61-s − 3·67-s + 3·79-s + 103-s + 109-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 2·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·7-s + 2·19-s + 25-s + 31-s − 37-s + 3·49-s + 3·61-s − 3·67-s + 3·79-s + 103-s + 109-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 2·175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.083676882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083676882\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.174562861719668583081566056735, −8.776972405607692961498789531765, −8.449167187839648686858355376303, −7.949497268168968776778115442177, −7.44581786522407746559651702457, −7.10559930711469195941974908517, −6.97267876167171196689573706985, −6.40173583459876422430965109846, −6.13543007974589541612230947422, −5.78013374214421030944250611222, −5.13228111935836079284942763387, −5.07106696181931412340056314717, −4.39990262138675167545200995909, −3.79486441903679941222616225541, −3.32940879611635655897020657269, −3.30580490963764900561089533974, −2.62716491819529291105978448016, −2.28447647272153658522705068103, −1.26955899974993790443636464035, −0.71597861909949129954474183164,
0.71597861909949129954474183164, 1.26955899974993790443636464035, 2.28447647272153658522705068103, 2.62716491819529291105978448016, 3.30580490963764900561089533974, 3.32940879611635655897020657269, 3.79486441903679941222616225541, 4.39990262138675167545200995909, 5.07106696181931412340056314717, 5.13228111935836079284942763387, 5.78013374214421030944250611222, 6.13543007974589541612230947422, 6.40173583459876422430965109846, 6.97267876167171196689573706985, 7.10559930711469195941974908517, 7.44581786522407746559651702457, 7.949497268168968776778115442177, 8.449167187839648686858355376303, 8.776972405607692961498789531765, 9.174562861719668583081566056735
