| L(s) = 1 | + 3-s + 2·4-s + 9-s + 2·12-s − 13-s − 12·23-s − 25-s + 27-s − 6·29-s + 2·36-s − 39-s + 4·43-s − 49-s − 2·52-s + 18·53-s + 19·61-s − 8·64-s − 12·69-s − 75-s − 2·79-s + 81-s − 6·87-s − 24·92-s − 2·100-s + 12·101-s − 23·103-s − 18·107-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 4-s + 1/3·9-s + 0.577·12-s − 0.277·13-s − 2.50·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1/3·36-s − 0.160·39-s + 0.609·43-s − 1/7·49-s − 0.277·52-s + 2.47·53-s + 2.43·61-s − 64-s − 1.44·69-s − 0.115·75-s − 0.225·79-s + 1/9·81-s − 0.643·87-s − 2.50·92-s − 1/5·100-s + 1.19·101-s − 2.26·103-s − 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484438475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484438475\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 13 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 109 T^{2} + p^{2} T^{4} \) |
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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
−11.37572848742358481212930607005, −10.59823767347400087322144830749, −10.13321637226882084185903141780, −9.667890877694940269195361998789, −9.006230570739869869063621724172, −8.301274914497699318907643390306, −7.84324938966090287359426718606, −7.20462745894953721189759593852, −6.75220340671584658064117429812, −5.94017406732674154585649666187, −5.44098834950524953712177500242, −4.22447691204362394779847142295, −3.73410968082917991436897373920, −2.52649427823520472776108712260, −1.97790661264898459429136024655,
1.97790661264898459429136024655, 2.52649427823520472776108712260, 3.73410968082917991436897373920, 4.22447691204362394779847142295, 5.44098834950524953712177500242, 5.94017406732674154585649666187, 6.75220340671584658064117429812, 7.20462745894953721189759593852, 7.84324938966090287359426718606, 8.301274914497699318907643390306, 9.006230570739869869063621724172, 9.667890877694940269195361998789, 10.13321637226882084185903141780, 10.59823767347400087322144830749, 11.37572848742358481212930607005
