| L(s) = 1 | + 2-s − 3·5-s − 7-s − 8-s − 3·10-s + 3·11-s − 2·13-s − 14-s − 16-s − 3·17-s + 7·19-s + 3·22-s + 9·23-s + 5·25-s − 2·26-s + 6·29-s − 8·31-s − 3·34-s + 3·35-s + 37-s + 7·38-s + 3·40-s + 6·41-s − 2·43-s + 9·46-s − 6·49-s + 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·5-s − 0.377·7-s − 0.353·8-s − 0.948·10-s + 0.904·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s + 1.60·19-s + 0.639·22-s + 1.87·23-s + 25-s − 0.392·26-s + 1.11·29-s − 1.43·31-s − 0.514·34-s + 0.507·35-s + 0.164·37-s + 1.13·38-s + 0.474·40-s + 0.937·41-s − 0.304·43-s + 1.32·46-s − 6/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.885595855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.885595855\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.763371079700271750033845065730, −9.698159435431574631186755687159, −9.022242203062570169394821173371, −8.994483356206098040072409811059, −8.402216170363493012973867422393, −7.73850861012365371308687606563, −7.62880991567855686257310721311, −7.03176463761525780201726242480, −6.63105713034330587579294960008, −6.49296630974517161359268483236, −5.59403279720130216399926511526, −5.18042310440136433883196151622, −4.88888309057666382372367342116, −4.25072262521787065918735842529, −3.98369057993054870699753205744, −3.23075419705373495479617535382, −3.18771199687414758820583624022, −2.43325876126771577361709349264, −1.36262377345782059353876066622, −0.58518293903836423228401425110,
0.58518293903836423228401425110, 1.36262377345782059353876066622, 2.43325876126771577361709349264, 3.18771199687414758820583624022, 3.23075419705373495479617535382, 3.98369057993054870699753205744, 4.25072262521787065918735842529, 4.88888309057666382372367342116, 5.18042310440136433883196151622, 5.59403279720130216399926511526, 6.49296630974517161359268483236, 6.63105713034330587579294960008, 7.03176463761525780201726242480, 7.62880991567855686257310721311, 7.73850861012365371308687606563, 8.402216170363493012973867422393, 8.994483356206098040072409811059, 9.022242203062570169394821173371, 9.698159435431574631186755687159, 9.763371079700271750033845065730
