| L(s) = 1 | − 2·7-s − 2·9-s − 5·17-s + 4·25-s + 4·31-s − 14·41-s + 2·47-s − 7·49-s + 4·63-s − 15·71-s + 8·73-s − 6·79-s − 5·81-s − 13·89-s − 35·97-s − 17·103-s + 19·113-s + 10·119-s − 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 10·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 2/3·9-s − 1.21·17-s + 4/5·25-s + 0.718·31-s − 2.18·41-s + 0.291·47-s − 49-s + 0.503·63-s − 1.78·71-s + 0.936·73-s − 0.675·79-s − 5/9·81-s − 1.37·89-s − 3.55·97-s − 1.67·103-s + 1.78·113-s + 0.916·119-s − 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.808·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 367 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 17 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 17 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−9.315708188935716743906341510994, −8.888793580610106524967975248679, −8.349505688281597729105260023012, −8.138429937812899929462182891586, −7.14913702652635708301086315421, −6.80504655959129523385635360065, −6.43646638185986515940980642145, −5.76336889502779327980365590399, −5.21504483465724654754420419149, −4.55870240969350431844198452995, −3.96455251478239306313953112113, −3.06025823301366601962549662004, −2.74559903564233577878691033622, −1.61933529819606053041583731875, 0,
1.61933529819606053041583731875, 2.74559903564233577878691033622, 3.06025823301366601962549662004, 3.96455251478239306313953112113, 4.55870240969350431844198452995, 5.21504483465724654754420419149, 5.76336889502779327980365590399, 6.43646638185986515940980642145, 6.80504655959129523385635360065, 7.14913702652635708301086315421, 8.138429937812899929462182891586, 8.349505688281597729105260023012, 8.888793580610106524967975248679, 9.315708188935716743906341510994
