| L(s) = 1 | − 2-s − 2·3-s − 2·4-s + 2·6-s + 4·7-s + 3·8-s + 3·9-s − 6·11-s + 4·12-s + 2·13-s − 4·14-s + 16-s + 4·17-s − 3·18-s − 8·21-s + 6·22-s − 13·23-s − 6·24-s − 2·26-s − 4·27-s − 8·28-s − 5·29-s + 4·31-s − 2·32-s + 12·33-s − 4·34-s − 6·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s + 1.51·7-s + 1.06·8-s + 9-s − 1.80·11-s + 1.15·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.707·18-s − 1.74·21-s + 1.27·22-s − 2.71·23-s − 1.22·24-s − 0.392·26-s − 0.769·27-s − 1.51·28-s − 0.928·29-s + 0.718·31-s − 0.353·32-s + 2.08·33-s − 0.685·34-s − 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3515625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3515625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 13 T + 87 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_4$ | \( 1 + 21 T + 191 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 17 T + 157 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 183 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.861677389970138395608394673370, −8.623243488503016890669362104948, −8.166569710710947018397642450711, −7.922690077848706067103992412508, −7.59955912095787795792560832279, −7.44728810778287889980142653029, −6.58991687120145699479680631188, −6.04867297392070441734572508289, −5.65594596680013671733621745766, −5.52836708552441876082696766650, −4.82197002913100883504934516817, −4.79664998334316709565876199899, −4.04496467352287442478388051716, −3.99519401965184691281326075170, −3.00435469623499470853923001781, −2.35690279268179068067664243479, −1.51733405099618903880292169801, −1.31547412610837567590861068193, 0, 0,
1.31547412610837567590861068193, 1.51733405099618903880292169801, 2.35690279268179068067664243479, 3.00435469623499470853923001781, 3.99519401965184691281326075170, 4.04496467352287442478388051716, 4.79664998334316709565876199899, 4.82197002913100883504934516817, 5.52836708552441876082696766650, 5.65594596680013671733621745766, 6.04867297392070441734572508289, 6.58991687120145699479680631188, 7.44728810778287889980142653029, 7.59955912095787795792560832279, 7.922690077848706067103992412508, 8.166569710710947018397642450711, 8.623243488503016890669362104948, 8.861677389970138395608394673370
