| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s + 7-s − 4·8-s + 8·10-s − 4·11-s − 13-s − 2·14-s + 8·16-s − 19-s − 8·20-s + 8·22-s + 2·25-s + 2·26-s + 2·28-s − 4·29-s − 9·31-s − 8·32-s − 4·35-s − 3·37-s + 2·38-s + 16·40-s + 10·41-s − 5·43-s − 8·44-s + 6·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s + 0.377·7-s − 1.41·8-s + 2.52·10-s − 1.20·11-s − 0.277·13-s − 0.534·14-s + 2·16-s − 0.229·19-s − 1.78·20-s + 1.70·22-s + 2/5·25-s + 0.392·26-s + 0.377·28-s − 0.742·29-s − 1.61·31-s − 1.41·32-s − 0.676·35-s − 0.493·37-s + 0.324·38-s + 2.52·40-s + 1.56·41-s − 0.762·43-s − 1.20·44-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.37222642487915167172257009170, −10.20154545275267843521912857988, −9.452465025961946570908891691014, −9.213899975782115287888013654052, −8.725951474637966049096781970938, −8.267641655680257937101482213602, −7.77381549174498013489541790435, −7.71617279529698604206832015639, −7.35823673067791072134822941807, −6.63223917867734229564889109429, −6.04693075638489360796595800883, −5.39067395617988889123664297475, −5.06178128445419203738211150598, −4.03184772875886666182640647734, −3.86954679327999317086999019538, −3.04680140792888561836748124065, −2.49011924522786841426748992925, −1.48237892803654071173364124564, 0, 0,
1.48237892803654071173364124564, 2.49011924522786841426748992925, 3.04680140792888561836748124065, 3.86954679327999317086999019538, 4.03184772875886666182640647734, 5.06178128445419203738211150598, 5.39067395617988889123664297475, 6.04693075638489360796595800883, 6.63223917867734229564889109429, 7.35823673067791072134822941807, 7.71617279529698604206832015639, 7.77381549174498013489541790435, 8.267641655680257937101482213602, 8.725951474637966049096781970938, 9.213899975782115287888013654052, 9.452465025961946570908891691014, 10.20154545275267843521912857988, 10.37222642487915167172257009170
