| L(s) = 1 | − 2-s + 5-s − 7-s + 8-s − 10-s − 3·11-s − 2·13-s + 14-s − 16-s − 2·17-s + 3·22-s + 6·23-s + 5·25-s + 2·26-s − 18·29-s − 5·31-s + 2·34-s − 35-s + 8·37-s + 40-s + 8·41-s − 6·46-s − 6·49-s − 5·50-s + 53-s − 3·55-s − 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.639·22-s + 1.25·23-s + 25-s + 0.392·26-s − 3.34·29-s − 0.898·31-s + 0.342·34-s − 0.169·35-s + 1.31·37-s + 0.158·40-s + 1.24·41-s − 0.884·46-s − 6/7·49-s − 0.707·50-s + 0.137·53-s − 0.404·55-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6651341297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6651341297\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
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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
−9.702105165812101155340764741377, −9.228874622860571233963296136863, −8.957166727655013590228035122803, −8.519454803906956597952295204182, −7.81727783062680140098936810632, −7.70922226890464415024628716616, −7.31441199253012094580805156215, −6.86243166876500771757020291510, −6.51781065145169803465985070499, −5.80321423332232422781415973908, −5.47591931937958104105076798934, −5.29982603943741196726822453912, −4.44806976801451086117434874471, −4.35328177653534753248689624179, −3.47980564258304087822140453607, −3.08564497086983222318166984851, −2.45865423644873072401007063510, −2.02187929730631216995548536627, −1.31565200169287834928097884951, −0.36302948413497093084514700695,
0.36302948413497093084514700695, 1.31565200169287834928097884951, 2.02187929730631216995548536627, 2.45865423644873072401007063510, 3.08564497086983222318166984851, 3.47980564258304087822140453607, 4.35328177653534753248689624179, 4.44806976801451086117434874471, 5.29982603943741196726822453912, 5.47591931937958104105076798934, 5.80321423332232422781415973908, 6.51781065145169803465985070499, 6.86243166876500771757020291510, 7.31441199253012094580805156215, 7.70922226890464415024628716616, 7.81727783062680140098936810632, 8.519454803906956597952295204182, 8.957166727655013590228035122803, 9.228874622860571233963296136863, 9.702105165812101155340764741377
