| L(s) = 1 | − 2-s − 2·3-s − 2·4-s + 2·6-s + 3·8-s + 2·9-s + 4·11-s + 4·12-s − 2·13-s + 16-s − 4·17-s − 2·18-s − 4·22-s − 8·23-s − 6·24-s + 2·26-s − 6·27-s + 10·29-s + 6·31-s − 2·32-s − 8·33-s + 4·34-s − 4·36-s − 6·37-s + 4·39-s − 14·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s + 1.06·8-s + 2/3·9-s + 1.20·11-s + 1.15·12-s − 0.554·13-s + 1/4·16-s − 0.970·17-s − 0.471·18-s − 0.852·22-s − 1.66·23-s − 1.22·24-s + 0.392·26-s − 1.15·27-s + 1.85·29-s + 1.07·31-s − 0.353·32-s − 1.39·33-s + 0.685·34-s − 2/3·36-s − 0.986·37-s + 0.640·39-s − 2.18·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 133 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 101 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 262 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 122 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 30 T + 398 T^{2} + 30 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 198 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.761063150317308201526623951396, −9.163044119562973724160734064967, −8.529683823137500537811703065931, −8.476369265678107975394008411230, −8.169168690899199613572833817714, −7.47547604306337529414069111501, −6.93058968201690610078986251254, −6.46313891296476344243431476870, −6.43070137101429226617203843308, −5.74806949845994802134984751338, −5.17321536099051064306185975283, −4.81951723250378498449505424031, −4.39517435935058781015767495428, −4.07374436810735628034842117309, −3.44006352602964746714616065249, −2.68106628449167227241616440923, −1.69992713057042086557034516787, −1.31710662654464286407497925571, 0, 0,
1.31710662654464286407497925571, 1.69992713057042086557034516787, 2.68106628449167227241616440923, 3.44006352602964746714616065249, 4.07374436810735628034842117309, 4.39517435935058781015767495428, 4.81951723250378498449505424031, 5.17321536099051064306185975283, 5.74806949845994802134984751338, 6.43070137101429226617203843308, 6.46313891296476344243431476870, 6.93058968201690610078986251254, 7.47547604306337529414069111501, 8.169168690899199613572833817714, 8.476369265678107975394008411230, 8.529683823137500537811703065931, 9.163044119562973724160734064967, 9.761063150317308201526623951396
