| L(s) = 1 | − 2·2-s − 7·5-s + 15·7-s + 8·8-s + 14·10-s − 15·11-s − 20·13-s − 30·14-s − 16·16-s − 16·17-s + 30·22-s − 6·23-s + 25·25-s + 40·26-s − 10·29-s − 93·31-s + 32·34-s − 105·35-s − 20·37-s − 56·40-s + 50·41-s − 30·43-s + 12·46-s − 150·47-s + 101·49-s − 50·50-s − 94·53-s + ⋯ |
| L(s) = 1 | − 2-s − 7/5·5-s + 15/7·7-s + 8-s + 7/5·10-s − 1.36·11-s − 1.53·13-s − 2.14·14-s − 16-s − 0.941·17-s + 1.36·22-s − 0.260·23-s + 25-s + 1.53·26-s − 0.344·29-s − 3·31-s + 0.941·34-s − 3·35-s − 0.540·37-s − 7/5·40-s + 1.21·41-s − 0.697·43-s + 6/23·46-s − 3.19·47-s + 2.06·49-s − 50-s − 1.77·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1346569940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1346569940\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T + 24 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 - 2 T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 15 T + 196 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T + 231 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 614 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 541 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T - 741 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 50 T + 819 T^{2} - 50 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 30 T + 2149 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 150 T + 9709 T^{2} + 150 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 60 T + 4681 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 64 T + 375 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 150 T + 11989 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 12 T + 6289 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 51 T + 7756 T^{2} + 51 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T - 8784 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.44752554642022450264470051480, −11.05191658672016625659453079852, −10.98951851292882810196410845382, −10.09874068106323329291497701339, −9.939553417988367157964915881886, −9.050573349438660352385943567428, −8.771706930941856070878398950574, −8.220517853170831652394505230525, −7.74530911999864929148722625643, −7.60225401199629046713308736193, −7.38137257526998623468403854405, −6.53617644359018449271900640239, −5.31336368989379202832091408637, −4.98405514151574655572178477055, −4.75993634009444587243692881853, −4.10063735077016094832242570920, −3.29011477193719489409813841697, −2.08814511690127314030002257411, −1.75707670620755864815347834653, −0.20069326056941450593496878120,
0.20069326056941450593496878120, 1.75707670620755864815347834653, 2.08814511690127314030002257411, 3.29011477193719489409813841697, 4.10063735077016094832242570920, 4.75993634009444587243692881853, 4.98405514151574655572178477055, 5.31336368989379202832091408637, 6.53617644359018449271900640239, 7.38137257526998623468403854405, 7.60225401199629046713308736193, 7.74530911999864929148722625643, 8.220517853170831652394505230525, 8.771706930941856070878398950574, 9.050573349438660352385943567428, 9.939553417988367157964915881886, 10.09874068106323329291497701339, 10.98951851292882810196410845382, 11.05191658672016625659453079852, 11.44752554642022450264470051480
