| L(s) = 1 | − 2-s + 2·3-s + 2·5-s − 2·6-s + 9·7-s + 3·9-s − 2·10-s − 11-s + 12·13-s − 9·14-s + 4·15-s − 7·17-s − 3·18-s − 19-s + 18·21-s + 22-s − 18·23-s + 5·25-s − 12·26-s − 16·29-s − 4·30-s − 2·31-s + 32-s − 2·33-s + 7·34-s + 18·35-s − 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 0.894·5-s − 0.816·6-s + 3.40·7-s + 9-s − 0.632·10-s − 0.301·11-s + 3.32·13-s − 2.40·14-s + 1.03·15-s − 1.69·17-s − 0.707·18-s − 0.229·19-s + 3.92·21-s + 0.213·22-s − 3.75·23-s + 25-s − 2.35·26-s − 2.97·29-s − 0.730·30-s − 0.359·31-s + 0.176·32-s − 0.348·33-s + 1.20·34-s + 3.04·35-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.458364119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.458364119\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 - 2 T + T^{2} + 4 T^{3} - 11 T^{4} + 4 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 2 T - T^{2} - 8 T^{3} + 41 T^{4} - 8 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - 9 T + 29 T^{2} - 33 T^{3} + 4 T^{4} - 33 p T^{5} + 29 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 12 T + 51 T^{2} - 76 T^{3} + 9 T^{4} - 76 p T^{5} + 51 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 7 T + 7 T^{2} - 115 T^{3} - 744 T^{4} - 115 p T^{5} + 7 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 9 T + 55 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 16 T + 67 T^{2} - 412 T^{3} - 4935 T^{4} - 412 p T^{5} + 67 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 2 T - 27 T^{2} - 116 T^{3} + 605 T^{4} - 116 p T^{5} - 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + 4 T - 21 T^{2} - 232 T^{3} - 151 T^{4} - 232 p T^{5} - 21 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 2 T + 23 T^{2} + 46 T^{3} + 665 T^{4} + 46 p T^{5} + 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 7 T + 67 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 37 T^{2} - 210 T^{3} + 1999 T^{4} - 210 p T^{5} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 12 T + 11 T^{2} + 654 T^{3} - 6491 T^{4} + 654 p T^{5} + 11 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 2 T + 65 T^{2} + 152 T^{3} + 4709 T^{4} + 152 p T^{5} + 65 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 23 T + 243 T^{2} + 2161 T^{3} + 18680 T^{4} + 2161 p T^{5} + 243 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 10 T + 154 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 24 T + 185 T^{2} - 456 T^{3} + 49 T^{4} - 456 p T^{5} + 185 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - 14 T + 123 T^{2} - 700 T^{3} + 821 T^{4} - 700 p T^{5} + 123 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 18 T + 65 T^{2} + 42 T^{3} + 1849 T^{4} + 42 p T^{5} + 65 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 26 T + 373 T^{2} + 4570 T^{3} + 47931 T^{4} + 4570 p T^{5} + 373 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 254 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 2 T - 33 T^{2} + 830 T^{3} + 4361 T^{4} + 830 p T^{5} - 33 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.093781975588859108324934516506, −7.977599680916252134459120038520, −7.963888366366607151651566328179, −7.79033381213849726021429186433, −7.23487653529941524357698586948, −6.90799334665341064683220874788, −6.56221347873411026291673372828, −6.53063217404418710586333437155, −6.12086743482653085557087210220, −5.67165545256966001338407811505, −5.62085977439024149242964255414, −5.35201652597748484991408689131, −5.18921194857340329270528575339, −4.55977361686521505240658822173, −4.40856939543952486267972775354, −4.12668965013348972973821882914, −3.72469303087348185420765368918, −3.71011300161005432616438855563, −3.47535636387407131184010383135, −2.51051114997803138718570421131, −2.14963870579464030505250677736, −1.82940283674798730342952289288, −1.76828937003339794255804046403, −1.74210905738127679859352162734, −0.816497378725594445056197074180,
0.816497378725594445056197074180, 1.74210905738127679859352162734, 1.76828937003339794255804046403, 1.82940283674798730342952289288, 2.14963870579464030505250677736, 2.51051114997803138718570421131, 3.47535636387407131184010383135, 3.71011300161005432616438855563, 3.72469303087348185420765368918, 4.12668965013348972973821882914, 4.40856939543952486267972775354, 4.55977361686521505240658822173, 5.18921194857340329270528575339, 5.35201652597748484991408689131, 5.62085977439024149242964255414, 5.67165545256966001338407811505, 6.12086743482653085557087210220, 6.53063217404418710586333437155, 6.56221347873411026291673372828, 6.90799334665341064683220874788, 7.23487653529941524357698586948, 7.79033381213849726021429186433, 7.963888366366607151651566328179, 7.977599680916252134459120038520, 8.093781975588859108324934516506
