| L(s) = 1 | + 3-s + 5-s + 5·11-s + 15-s − 4·17-s − 8·19-s − 4·23-s + 5·25-s − 27-s − 10·29-s − 3·31-s + 5·33-s + 4·37-s − 4·43-s + 6·47-s − 4·51-s + 9·53-s + 5·55-s − 8·57-s + 11·59-s − 6·61-s − 2·67-s − 4·69-s − 4·71-s + 10·73-s + 5·75-s + 3·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.50·11-s + 0.258·15-s − 0.970·17-s − 1.83·19-s − 0.834·23-s + 25-s − 0.192·27-s − 1.85·29-s − 0.538·31-s + 0.870·33-s + 0.657·37-s − 0.609·43-s + 0.875·47-s − 0.560·51-s + 1.23·53-s + 0.674·55-s − 1.05·57-s + 1.43·59-s − 0.768·61-s − 0.244·67-s − 0.481·69-s − 0.474·71-s + 1.17·73-s + 0.577·75-s + 0.337·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.506189173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.506189173\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−9.264099594723781224880489550601, −8.674904681914837763667206653583, −8.623122995027749086326652796862, −8.210204336131024636060434124044, −7.62889102428376101164693166654, −7.07484583946968114303913920671, −6.98476033230961536257573073360, −6.49128971882048352704348200360, −5.97351072899392792649825708485, −5.90446912952814723134505483135, −5.25456993932562790526417908683, −4.67331392305461226897948667618, −4.24147627282244726410142343926, −3.80262749528093413368229290515, −3.73696130146704588161860501122, −2.81322368978783275942551703448, −2.38084421049568660477990229632, −1.89649472878622797180207758959, −1.53166094255051119392166571361, −0.49950331082421080325331240143,
0.49950331082421080325331240143, 1.53166094255051119392166571361, 1.89649472878622797180207758959, 2.38084421049568660477990229632, 2.81322368978783275942551703448, 3.73696130146704588161860501122, 3.80262749528093413368229290515, 4.24147627282244726410142343926, 4.67331392305461226897948667618, 5.25456993932562790526417908683, 5.90446912952814723134505483135, 5.97351072899392792649825708485, 6.49128971882048352704348200360, 6.98476033230961536257573073360, 7.07484583946968114303913920671, 7.62889102428376101164693166654, 8.210204336131024636060434124044, 8.623122995027749086326652796862, 8.674904681914837763667206653583, 9.264099594723781224880489550601
