| L(s) = 1 | + 3·3-s + 2·4-s − 7-s + 6·9-s + 6·12-s + 7·13-s − 7·19-s − 3·21-s − 5·25-s + 9·27-s − 2·28-s + 7·31-s + 12·36-s − 21·37-s + 21·39-s − 10·43-s − 6·49-s + 14·52-s − 21·57-s − 12·61-s − 6·63-s − 8·64-s + 21·67-s + 7·73-s − 15·75-s − 14·76-s − 13·79-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4-s − 0.377·7-s + 2·9-s + 1.73·12-s + 1.94·13-s − 1.60·19-s − 0.654·21-s − 25-s + 1.73·27-s − 0.377·28-s + 1.25·31-s + 2·36-s − 3.45·37-s + 3.36·39-s − 1.52·43-s − 6/7·49-s + 1.94·52-s − 2.78·57-s − 1.53·61-s − 0.755·63-s − 64-s + 2.56·67-s + 0.819·73-s − 1.73·75-s − 1.60·76-s − 1.46·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.252041185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.252041185\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−12.17086717762798092021319711217, −11.69669932473786528121947326696, −10.92550329199480167008489690258, −10.89788181295720255007777603854, −10.01809902762986195908210035193, −9.959398003452045865289801423866, −9.097450989864690586399106326145, −8.608829076573793060916093381791, −8.349732452444268304290543035431, −8.087841502792574714796864995698, −7.04112578121127861852040703844, −6.93883192950769298150807432880, −6.32528579230904434403694926178, −5.85499557552074227026409362878, −4.77241622397165056600171208375, −4.07425742122661719516366678293, −3.38298700081458132887613400183, −3.17114009101357850970073437250, −1.97989025242404553687679101681, −1.80058772273606642429886052316,
1.80058772273606642429886052316, 1.97989025242404553687679101681, 3.17114009101357850970073437250, 3.38298700081458132887613400183, 4.07425742122661719516366678293, 4.77241622397165056600171208375, 5.85499557552074227026409362878, 6.32528579230904434403694926178, 6.93883192950769298150807432880, 7.04112578121127861852040703844, 8.087841502792574714796864995698, 8.349732452444268304290543035431, 8.608829076573793060916093381791, 9.097450989864690586399106326145, 9.959398003452045865289801423866, 10.01809902762986195908210035193, 10.89788181295720255007777603854, 10.92550329199480167008489690258, 11.69669932473786528121947326696, 12.17086717762798092021319711217
