| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s − 3·9-s + 2·11-s − 8·13-s − 4·15-s − 4·17-s − 2·19-s + 4·21-s + 2·23-s + 3·25-s − 14·27-s − 10·29-s − 4·31-s + 4·33-s − 4·35-s − 6·37-s − 16·39-s − 6·41-s + 6·43-s + 6·45-s − 10·47-s − 6·49-s − 8·51-s − 12·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s − 9-s + 0.603·11-s − 2.21·13-s − 1.03·15-s − 0.970·17-s − 0.458·19-s + 0.872·21-s + 0.417·23-s + 3/5·25-s − 2.69·27-s − 1.85·29-s − 0.718·31-s + 0.696·33-s − 0.676·35-s − 0.986·37-s − 2.56·39-s − 0.937·41-s + 0.914·43-s + 0.894·45-s − 1.45·47-s − 6/7·49-s − 1.12·51-s − 1.64·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 153 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 22 T + 274 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.926218000403695387159332084639, −8.800483037629403963701859363745, −8.076597670308677639796445279006, −8.058178405358575091257182183553, −7.57556698387068727437587080638, −7.32807957413293584711682161824, −6.68710021124262154165815009113, −6.54200464943715388711901528322, −5.53410208310795347432459811298, −5.50440479705808773231151571596, −4.79450616879413474957441780601, −4.58506726505908972761776961770, −3.97305926689398884035415628360, −3.45180272862097588411624472908, −3.11583686011888288294257461115, −2.59321736160339043091915744112, −1.96535384695599674898545238778, −1.75398930023516601792799084058, 0, 0,
1.75398930023516601792799084058, 1.96535384695599674898545238778, 2.59321736160339043091915744112, 3.11583686011888288294257461115, 3.45180272862097588411624472908, 3.97305926689398884035415628360, 4.58506726505908972761776961770, 4.79450616879413474957441780601, 5.50440479705808773231151571596, 5.53410208310795347432459811298, 6.54200464943715388711901528322, 6.68710021124262154165815009113, 7.32807957413293584711682161824, 7.57556698387068727437587080638, 8.058178405358575091257182183553, 8.076597670308677639796445279006, 8.800483037629403963701859363745, 8.926218000403695387159332084639
