| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 2·5-s + 9·6-s − 4·7-s − 3·8-s + 2·9-s + 6·10-s − 5·11-s − 12·12-s + 5·13-s + 12·14-s + 6·15-s + 3·16-s + 17-s − 6·18-s + 2·19-s − 8·20-s + 12·21-s + 15·22-s + 11·23-s + 9·24-s − 2·25-s − 15·26-s + 6·27-s − 16·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 0.894·5-s + 3.67·6-s − 1.51·7-s − 1.06·8-s + 2/3·9-s + 1.89·10-s − 1.50·11-s − 3.46·12-s + 1.38·13-s + 3.20·14-s + 1.54·15-s + 3/4·16-s + 0.242·17-s − 1.41·18-s + 0.458·19-s − 1.78·20-s + 2.61·21-s + 3.19·22-s + 2.29·23-s + 1.83·24-s − 2/5·25-s − 2.94·26-s + 1.15·27-s − 3.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 43 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 31 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 11 T + 75 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 65 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 87 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 103 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 111 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 123 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 137 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 163 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T - 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 169 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 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
−9.153977517532590052273936750281, −8.727670441824332670059547046330, −8.249104312004390166682832740262, −8.048794747828001243482315470844, −7.56596489010481016075079183953, −7.12749552882590490191928079526, −6.84501209372292946358705588151, −6.21033964997663220814003065208, −6.03849565496797992453318616063, −5.62872462324428161697999258013, −5.10734865966266866981684334966, −4.78628003675537005557154935940, −3.94741568271863643085453702472, −3.44041233431509068387055370179, −2.90729373121675889681895518828, −2.60813248420782571482756066315, −1.05100824094410808861932932596, −1.03151024896456511366136557881, 0, 0,
1.03151024896456511366136557881, 1.05100824094410808861932932596, 2.60813248420782571482756066315, 2.90729373121675889681895518828, 3.44041233431509068387055370179, 3.94741568271863643085453702472, 4.78628003675537005557154935940, 5.10734865966266866981684334966, 5.62872462324428161697999258013, 6.03849565496797992453318616063, 6.21033964997663220814003065208, 6.84501209372292946358705588151, 7.12749552882590490191928079526, 7.56596489010481016075079183953, 8.048794747828001243482315470844, 8.249104312004390166682832740262, 8.727670441824332670059547046330, 9.153977517532590052273936750281
