| L(s) = 1 | − 2·2-s − 3·3-s − 4·5-s + 6·6-s + 8·8-s + 6·9-s + 8·10-s + 7·13-s + 12·15-s − 16·16-s + 32·17-s − 12·18-s + 38·19-s + 12·23-s − 24·24-s + 25·25-s − 14·26-s − 9·27-s − 52·29-s − 24·30-s − 64·34-s − 2·37-s − 76·38-s − 21·39-s − 32·40-s + 44·41-s + 69·43-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s − 4/5·5-s + 6-s + 8-s + 2/3·9-s + 4/5·10-s + 7/13·13-s + 4/5·15-s − 16-s + 1.88·17-s − 2/3·18-s + 2·19-s + 0.521·23-s − 24-s + 25-s − 0.538·26-s − 1/3·27-s − 1.79·29-s − 4/5·30-s − 1.88·34-s − 0.0540·37-s − 2·38-s − 0.538·39-s − 4/5·40-s + 1.07·41-s + 1.60·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7892821608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7892821608\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T - 9 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 95 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 230 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 7 T - 120 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 32 T + 735 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12 T + 577 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 52 T + 1863 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1775 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 44 T + 255 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 69 T + 3436 T^{2} - 69 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T + 4909 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 44 T - 873 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T + 4453 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T - 3600 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 87 T + 7012 T^{2} - 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T + 7069 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 5 T - 5304 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 129 T + 11788 T^{2} + 129 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3550 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T + 8979 T^{2} + 130 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 74 T - 3933 T^{2} + 74 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−12.31010402047882183475948593348, −11.56594059282975788134153328790, −11.06489481822866966071400738934, −11.01540048608220126794575172513, −10.14366429900187255098429540736, −9.994866054781794871208697106409, −9.139610228814041675378329111575, −9.088319858435315672318531018168, −8.237593159688291839615152817448, −7.62927007535048395747022564911, −7.29617779746713764712514367584, −7.13642469850524433409522506414, −5.91737525991885340216231190067, −5.44837082698884956172666089408, −5.20669779752266848630632772356, −3.94200922063773251294286226276, −3.91916637429854361691269997645, −2.71565411846103758334969329664, −1.11826455889140503244677097679, −0.808017667661350721789705766349,
0.808017667661350721789705766349, 1.11826455889140503244677097679, 2.71565411846103758334969329664, 3.91916637429854361691269997645, 3.94200922063773251294286226276, 5.20669779752266848630632772356, 5.44837082698884956172666089408, 5.91737525991885340216231190067, 7.13642469850524433409522506414, 7.29617779746713764712514367584, 7.62927007535048395747022564911, 8.237593159688291839615152817448, 9.088319858435315672318531018168, 9.139610228814041675378329111575, 9.994866054781794871208697106409, 10.14366429900187255098429540736, 11.01540048608220126794575172513, 11.06489481822866966071400738934, 11.56594059282975788134153328790, 12.31010402047882183475948593348
